On the volumes and affine types of trades

A $[t]$-trade is a pair $T=(T_+, T_-)$ of disjoint collections of subsets (blocks) of a $v$-set $V$ such that for every $0\le i\le t$, any $i$-subset of $V$ is included in the same number of blocks of $T_+$ and of $T_-$. It follows that $|T_+| = |T_-|$ and this common value is called the volume of $T$. If we restrict all the blocks to have the same size, we obtain the classical $t$-trades as a special case of $[t]$-trades. It is known that the minimum volume of a nonempty $[t]$-trade is $2^t$. Simple $[t]$-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most $v-t-1$. From the characterization of Kasami--Tokura of such functions with small number of ones, it is known that any simple $[t]$-trade of volume at most $2\cdot2^t$ belongs to one of two affine types, called Type\,(A) and Type\,(B) where Type\,(A) $[t]$-trades are known to exist. By considering the affine rank, we prove that $[t]$-trades of Type\,(B) do not exist. Further, we derive the spectrum of volumes of simple trades up to $2.5\cdot 2^t$, extending the known result for volumes less than $2\cdot 2^t$. We also give a characterization of"small"$[t]$-trades for $t=1,2$. Finally, an algorithm to produce $[t]$-trades for specified $t$, $v$ is given. The result of the implementation of the algorithm for $t\le4$, $v\le7$ is reported.


Introduction
Let v, k, t be positive integers such that v > k > t and V be a v-set. Suppose that T + and T − are two disjoint collections of k-subsets of V (called blocks) such that the occurrences of every t-subset of V in T + and T − are the same. Then T = (T + , T − ) is called a t-(v, k) trade (or a t-trade when the role of v, k is not important). Basically, t-trades have been defined and utilized in connection with t-designs: if D 1 and D 2 are two t-designs with the same parameters and the same ground set V , then (D 1 \ D 2 , D 2 \ D 1 ) is a t-trade. In this paper we consider [t]-trades, a generalization of t-trades, relaxed in the sense that the block size is not fixed. More precisely, a [t]-trade is a pair T = (T + , T − ) of disjoint collections of subsets of V such that for every 0 ≤ i ≤ t, every i-subset of V is included in the same number of blocks of T + and of T − . Note that any t-trade is also an i-trade for every 0 ≤ i ≤ t, which means that any t-trade is a [t]-trade as well. On the other hand, [t]-trades can be naturally treated as trades of orthogonal arrays: given two orthogonal binary arrays A 1 , A 2 with the same parameters and strength t, their difference pair (A 1 \A 2 , A 2 \A 1 ) is a [t]-trade (here, each array is treated as the set of its row-tuples).
For a [t]-trade T = (T + , T − ) we have |T + | = |T − | and this common value is called the volume of T and denoted by vol(T ). It is known that the smallest volume of a nonempty t-trade is 2 t which was determined independently in [5,6] and [2]. For the volumes (of t-trades) between 2 t and 2·2 t , it was conjectured by Khosrovshahi and Malik [10,14] and by Mahmoodian and Soltankhah [17,13] (see also [4]) that any volume in this range is of the form 2 t+1 − 2 i for some i ∈ {0, . . . , t − 1}. This was known as "the gaps conjecture" which was proved recently in [11] for simple trades (for the trades with repeated blocked, the problem remains open). We note that the spectrum of volumes of t-trades and that of [t]-trades are the same [11] (i.e., a t-trade of volume m exists if and only if a [t]-trade of volume m exists). This is a key observation which allows one to translate problems related to the volumes of t-trades to the setting of [t]-trades; the strategy which was employed in settling the gaps conjecture for simple trades [11]. Further important problems in design theory can be described in terms of volumes of trades. For instance, the celebrated halving conjecture [3] can be considered as a partial case of the problem of determining the maximum volume of t-(v, k) trades (which is conjectured to be 1 2 v k whenever v−i k−i is even for all i = 0, . . . , t). This is one of the motivations to study [t]-trades as a new tool to attack problems in combinatorial design theory which can be described in terms of (volumes) of t-trades.
In this paper we further study [t]-trades and their volumes. As noted in [11], any simple (i.e., with no repeated blocks) [t]-trade corresponds to a Boolean function of degree at most v − t − 1 (where v is the number of arguments). From the characterization of such functions with small number of ones (given in [7]), it is observed that any simple [t]-trade of volume at most 2 · 2 t belongs to one of the two affine types, called Type (A) and Type (B) (Type (A) [t]-trades are known to exist). Existence of [t]-trades of Type (B) was declared as an open problem in [11]. By considering the affine rank, we prove that [t]-trades of Type (B) do not exist. Also from our results on affine rank of trades, we derive the spectrum of volumes of trades up to 2.5 · 2 t extending the gaps conjecture proved in [11].
The paper is organized as follows. Section 2 contains main definitions. In Section 3 we prove some auxiliary statements. In Section 4, we consider the affine rank of simple [t]-trades. We utilize these considerations to prove the non-existence of simple [t]-trades of Type (B) as well as simple [t]-trades of volume 2 t+1 + 2 i , (t − 1)/2 ≤ i ≤ t − 4. Based on this latter non-existential result and the construction of [t]-trades of volumes 2 t+1 + 2 t−1 − 2 i , 0 ≤ i ≤ t − 2, and 2 t+1 + 2 t−1 − 3 · 2 i , 0 ≤ i ≤ t − 3, in Section 5 we characterize the spectrum of volumes of simple [t]-trades up to the value 2.5 · 2 t exclusively. Section 6 is devoted to the characterization of [1]-trades of volume 3 and [2]-trades of volume 6. Section 7 contains the results of an exhaustive computer enumeration of the equivalence classes of trades for small t, and small foundations and volumes.
Finally, we note that our results are applicable to the classical t-trades. Indeed, on one hand, the t-trades are a special case of the [t]-trades; on the other hand, every [t]-trade can be mapped to a t-trade with a fixed block size by some affine transformation [11]. However, the characterization results for [t]-trades do not imply that the corresponding t-trades are also characterized up to isomorphism. Indeed, the class of equivalence transformations for [t]-trades is larger than that of t-trades (it contains shifts), and nonisomorphic t-trades could be equivalent as [t]-trades. As an example of the characterization of small t-trades, we mention the classification in [1, Table 3.4] of the Steiner 2-trades with block size 3, volume at most 9 and foundation size at most 11, where the additional "Steiner" property means that no pair of elements is included in more than one block of each leg of the trade.

[t]-trades
Let t, v be positive integers with t < v. The subsets of V = {1, . . . , v} will be associated with their characteristic v-tuples, e.g., {2, 3, 6} = (0, 1, 1, 0, 0, 1, 0) = 0110010 for v = 7. The cardinality of a subset (the number of 1's in the corresponding tuple) will be referred to as its size. The set of all subsets of V is denoted by 2 V , which forms a group isomorphic to Z v 2 , with the symmetric difference as the group operation. The symmetric difference corresponds to the bitwise modulo-2 addition of the characteristic v-tuples, and we will use ⊕ as the symbol for this operation. In many cases, we will omit this symbol, i.e., XY := X ⊕ Y . For every i ∈ V , we denote x i := {i}. Therefore, every By a [t]-trade we mean a pair T = (T + , T − ) of disjoint collections of 2 V such that for every i ∈ [t], [t] := {0, . . . , t}, every i-subset of V is included in the same number of elements of T + and of T − . The sets T + and T − are called the legs of T and the elements of T + and T − are referred to as the blocks of T . A trade is called simple if it has no repeated blocks; in that case, T + and T − can be considered as ordinary sets. The cardinality of a leg (which is, trivially, the same for both legs) is called the volume of T , denoted by vol(T ). The foundation of T , denoted by found(T ), is the set of all ℓ ∈ V such that ℓ appears in some blocks of T . For any ℓ ∈ found(T ), the replication of ℓ is defined as We use the same notation for the subsets α ⊂ found(T ) with |α| ≤ t: The trade of volume 0 is called void.
An element ℓ is said to be essential for a trade T if T has a block containing ℓ and a block not containing ℓ.
A trade can be treated as a Z-valued function over 2 V , and written as where the positive coefficients τ X equal the multiplicity of X in T + , and the negative coefficients τ X equal minus the multiplicity of X in T − . In terms of such functions, (1), the definition of a [t]-trade can be rewritten as X⊇S τ X = 0, for every S ∈ 2 V such that |S| ≤ t.
Below, we formally consider summation and multiplication of functions in form (1), using the rules of the group ring Z[(2 V , ⊕)]. This language is convenient for the representation of the trades of small volumes.
A subset T of 2 V is said to be a [t]-unitrade if for every subset S of V with |S| ≤ t, the number of blocks of T including S is an even number. A [t]-unitrade has necessarily an even number of blocks. If (T + , T − ) is a simple [t]-trade, then clearly T + ∪ T − is a [t]-unitrade. We extend the definition of volume, foundation and replication to include unitrades T by vol(T ) := |T |/2, r ℓ := |{B ∈ T : ℓ ∈ B}|/2, and similarly for subsets of found(T ).

The binary vector space, Boolean functions and polynomials
The set 2 V with the addition operation ⊕ and the natural scalar multiplication by 0 and 1 is a v-dimensional vector space over the Galois field GF(2) = ({0, 1}, ⊕). Every subset S of 2 V can be represented by the characteristic {0, 1}-function over 2 V (such functions are known as Boolean functions), which, in turn, is uniquely represented as a polynomial of degree at most v in the vector coordinates y 1 , . . . , y v in the standard basis x 1 = {1}, . . . , x v = {v}, over GF (2). We will say that this polynomial is associated with the set S.
The set of all {0, 1}-functions on 2 V represented by polynomials of degree at most m is denoted by RM(m, v) (in coding theory, this is known as the Reed-Muller code of order m).

Preliminary lemmas
In this section we establish some basic facts about [t]-trades which will be used in the rest of the paper. We start with a result which reveals the connection between [t]-trades and Reed-Muller codes. Consider f (y 1 , . . . , y 6 ) = y 1 y 2 y 3 + y 1 y 2 y 4 ∈ RM (3,6). The set of ones of f is T = {111000, 111001, 111010, 111011, 110100, 110101, 110110, 110111}.
It is easily seen that T is indeed a [2]-unitrade. This is an example of the following general fact. Proof. We divide the argument in three parts.
(i) Consider a monomial f = y i1 · · · y i ℓ , and let T be the set of ones of f . Given a subset S of V , we count the number of the members of T 'including' S (in terms of tuples, having 1's in all positions from S). For a binary vector a = a 1 . . . a v , we have f (a) = 1 and a includes S if and only if a i = 1 for all i ∈ S ∪ {i 1 , . . . , i ℓ }. So the number of members of T including S is 2 |V \(S∪{i1,...,i ℓ })| . This number is even if and only if V \ (S ∪ {i 1 , . . . , i ℓ }) is nonempty.
(ii) In particular, if l ≤ m, then 2 |V \(S∪{i1,...,i ℓ })| is even for every S of size |S| ≤ t = v − m − 1. So, for every monomial of degree less than v − t, the associated set is a [t]-unitrade. This extends to every polynomial of degree less than v − t (i.e., at most m), because any linear combination over GF(2) preserves the parity properties defining a [t]-unitrade.
(iii) On the other hand, if the degree s of a polynomial is v − t or more, then it includes some monomial y i1 · · · y is with coefficient 1 and does not meet the definition of a [t]-unitrade with S = V \{i 1 , . . . , i s }, |S| ≤ t. Indeed, by the 'only if' statement of (i), for this monomial, the set T of ones has odd number of elements including S; on the other hand, for every other monomial of degree at most s this number is even, by the 'if' statement of (i); hence, for the whole polynomial, it is odd.
In view of Lemma 2, the next claim is just the well-known fact on Hamming distance of RM(m, v) (see, e.g., [12,Theorem 3 in 13.3]), which is easy to prove by induction on t.
The same bound holds for [t]-trades. The following lemma gives the structure of [t]-trades with the minimum volume. A version of this result for t-trades is quite well-known, but it can be easily generalized to [t]-trades.
For Y ∈ 2 V and a function T : 2 V → Z, we call Y T the Y -shift, or simply a shift of T .
Example 1. The function Given a trade T in the form (1) and an element i ∈ V , by the i-projection, or simply a projection, of T we mean the function T i obtained from T by removing i from every block that contains i. Hence, T i = P + P ′ , where T = P + x i P ′ and i does not occur in P and P ′ .
Note that after a projection, it is possible that two blocks cancel out each other, so the volume can be reduced. If the volume of T equals the volume of T i , then we say that T is an extension of T i . So, an extension of a [t]-trade T is a [t]-trade obtained from T by including a new element in some blocks of T .
The following four lemmas are straightforward from the definitions.

Lemma 6. A projection of a [t]-trade is a [t]-trade.
Lemma 7. Let T = P + x i P ′ be a [t]-trade, where i does not occur in the blocks of P , P ′ . Then P , P ′ , and x i P ′ are [t − 1]-trades.
Lemma 9. If P is a [t − 1]-trade and the element i does not occur in its blocks, then We say that a [t]-trade is s-small for some s > 1 if its volume is less than s · 2 t . The 2-small trades will be referred to as small.
The following statement plays an important role in the computer-aided classification of small [t]-trades.
where T i is a [t]-trade, P and P ′ are [t − 1]-trades, and the element i does not occur in Proof. If we present the [t]-trade in the form T = P + x i P ′ and define T i = P + P ′ to be the i-projection of T , then the first statement trivially follows from Lemmas 7 and 6. The volume of the projection is trivially not greater than the volume of the original trade; so, if T is s-small then so is T i . Moreover, the volume of T is the sum of the volumes of P and P ′ ; so, if it is less than s · 2 t , then one of the summands is less than s · 2 t−1 , which means that the corresponding [t − 1]-trade is s-small.
As mentioned before, the minimum distance of RM(m, v) is d = 2 v−m . Kasami and Tokura [7] characterized codewords of RM(m, v) with weight at most 2d. This result is the base of our characterization of [t]-trades with small volumes.

Lemma 11 ([7]).
Any Boolean function f from RM(m, v) of weight greater than 2 v−m and less than 2 · 2 v−m can be reduced by an invertible affine transformation of its variables to one of the following forms: Based on Lemma 2 and the Kasami-Tokura characterization, the gaps conjecture was proved in [11] in the more general setting of [t]-unitrades. For future reference, we state it as the following lemma.
In particular, the same holds for simple [t]-trades.
and so it has an even cardinality. Considering the vectors of A as subsets of V , this means that {i 1 , . . . , i r } is contained in an even number of blocks of A.
Proof. Let d be the dimension of T . By Lemma 12, |T | ≥ 2 t+1 . Therefore, d ≥ t + 1, and hence by Lemma 13, T is a [t]-unitrade. It follows that T \ T is also a [t]-unitrade.
We denote the trade (R + , R − ) of Lemma 15 by T αβ . In particular, we use the notation T i for α = {i} and β = ∅ and T j for α = ∅ and β = {j}.

We call a [t]-trade T reduced if
for all i ∈ found(T ).

Lemma 16. Every [t]-trade can be transformed by some shifts into a reduced [t]-trade.
Proof. Let T be a [t]-trade, and let I consist of all i's such that r i > 1 2 vol(T ). In I ⊕ T , the I-shift of T , the replication of i is vol(T ) − r i < 1 2 vol(T ) for every i ∈ I (the replications of elements in V \ I remains the same). It follows that I ⊕ T is reduced.

Affine rank of simple [t]-trades
Recall that by Lemma 2, unsigned simple [t]-trades with a foundation of size v can be regraded as codewords of the Reed-Muller code RM(v − t − 1, v). As given in Lemma 11, the codewords of Reed-Muller codes with weights at most twice the minimum distance have been characterized in [7] and subsequently divided into Types (A) or (B). Accordingly, simple [t]-trades (and also [t]-unitrades) with volume at most 2 t+1 can be categorized into Types (A) or (B). Krotov [11] considered this possible dichotomy and put forward the existence of [t]-trades of Type (B) as an open problem. In this section we establish some results about the affine rank of trades from which it follows that trades of Type (B) do not exist. In addition, the non-existence of simple [t]-trades with volumes 2 t+1 We denote the affine rank (the dimension of the affine span) of a subset S of the vector space We first show how the types of [t]-trades can be distinguished by means of their affine rank.
Proof. Let T ′ denote the corresponding [t]-unitrade with T . Note that an invertible affine transformation of the variables does not change the affine rank and the cardinality of the set of ones of the polynomials given in Lemma 11. So we may assume that T ′ is the set of ones of such polynomials.
(i) Considering the associated polynomial of T ′ given by Lemma 11 (A), it is seen that T ′ is the symmetric difference of two intersecting affine subspaces of dimension t + 1. If the dimension of the intersection is i, 0 ≤ i < t, then the cardinality of T ′ is 2 t+2 − 2 i+1 and its affine rank is 2t + 2 − i.
(ii) T ′ is the set of ones of the polynomial given by Lemma 11 (B). By a counting argument, we have A unitrade of Type (B) is an intersection of an affine subspace of dimension t + 3 and the set of ones of a quadratic function. So afrk( From Lemma 11 it is clear that [t]-unitrades of Type (B) (and so with affine rank t + 3) do exist. However, we manage to prove that this is not the case for [t]-trades. It follows that unitrades of Type (B) are not 'splittable.' This means that, although an unsigned [t]-trade gives a [t]-unitrades, but this is not reversible in general.
Proof. Suppose that vol(T ) > 2 t . So by Lemma 12,vol -trade of minimum volume and t ≥ 3, there exists some k ∈ found(T ) with r ijk = 2 t−3 . It turns out that r ik , r jk ∈ {0, 2 t−1 } and so r ik = r jk = 2 t−2 . Then It follows that T ijk has affine rank at least t + 1. On the other hand, as vol(T ijk ) = vol(T jik ) = vol(T kij ) = 2 t−3 = 0, there are three more affinely independent vectors in T each containing exactly one of i, j or k. This means that the affine rank of T is at least t + 4.
and afrk(T ) = t+3, then the associated polynomial corresponding to T can be obtained from the associated polynomial to T is of the form (5) with a = 1, and afrk(T ) = t + 3.
Proof. (i) If afrk(T ) = t + 3, then there is an invertible affine variable transformation that sends T to a [t]-unitrade T ′ whose affine span is defined by the equations It follows from (6) that the polynomial associated to T ′ has the form By Lemma 2, g has degree at most m, and hence h has degree at most 2. The polynomial h, as a polynomial in the t + 3 variables y m−1 , . . . , y v , has 2vol(T ) ones, which is either 2 t+2 − 2 i+1 or 2 t+2 + 2 i+1 . By the results of [16], h is affinely equivalent to with a = 0 or a = 1, respectively. Therefore, g is affinely equivalent to f in (5).
Proof. By Lemma 19, the associated polynomial corresponding to T can be obtained from (5) by an invertible affine transformation of variables. We have So y v is a free variable of f , which implies that r v = vol(T )/2. In fact the set of ones of f is of the form S × {0, 1} for some S ⊂ 2 [v−1] with |S| = vol(T ). Let y → yM + b be the invertible affine transformation which gives the associated polynomial of T . Hence T is the set of ones of The last row of M −1 should be nonzero. So we may assume that the j-th column of M −1 , say a ⊤ has its last component equal to 1. Then we have either Proof. Denote by A the affine span of T , and by A i , the i-projection of A. If |A i | < |A| for all i ∈ found(T ), then A = 2 found(T ) , and the statement trivially holds with T ′ = T . Otherwise, |A i | = |A| for some i ∈ found(T ), and the i-projecting acts bijectively on A. It follows that the i-projection of T has the same volume and affine rank as T , but smaller foundation. Repeating this operation |found(T )| − afrk(T ) times, we find a required T ′ .  The proof of Lemma 22 is by computation and will be addressed in Section 7. The sharpening claims in the parenthesis can be easily shown theoretically, but we will not use them in the further discussion.
Proof. As shifts do not change the volume and affine rank of trades, in view of Lemma 16, we may assume that T is a reduced simple [t]-trade.
Suppose vol(T ) = 14. For a contradiction, let afrk(T ) = 6. By Lemma 21, we may assume that |found(T )| = 6. Applying Lemma 12 to T i we obtain r i ∈ {4, 6, 7} for all i ∈ found(T ). If r i = 7 for some i ∈ found(T ), then T i is a [2]-trade of volume 7 and has affine rank at least 6 by Proposition 17. Hence afrk(T ) ≥ 7, a contradiction. Hence for all i ∈ found(T ), r i = 4 or 6. If for all i ∈ found(T ), r i = 4, then we are done by Lemma 18. So assume that r i = 6 for some i ∈ found(T ). Here T i is a [2]-trade of volume 6 and |found(T i )| = 5 (|found(T i )| cannot be smaller than 5 as afrk(T i ) = 5). Note that vol(T i ) = 8. Also |found(T i )| = 5, because T ij is a [1]-trade and so afrk(T ij ) ≥ 4, it follows that afrk(T i ) ≥ 5. On the other hand, afrk(T i ) ≤ afrk(T ) − 1 = 5. Our aim is to obtain a contradiction by considering the replications of elements in both T i and T i . In view of Lemma 22 applied to T i , the number of j ∈ found(T ) with r ij = 3 must be odd. We further claim that r ij = 3 if and only if r ij = 3. The claim follows from the fact that if either r ij = 3 or r ij = 3, then r j = 6; since otherwise, r j = 4, and then T ij or T ij would be a [1]-trade of volume 1, a contradiction. Also there are no k ∈ found(T ) with r ik = 5; since otherwise r k is necessarily 6, and so T ik would be a [1]-trade of volume 1, again a contradiction. The above argument shows that the number of elements with an odd replication in T i is the same as the number of elements with an odd replication in T i . However, by Lemma 22, the former is an odd number and the latter is an even number, again a contradiction.
We claim that r k = 8 for some k ∈ found(T ). Otherwise, r i ∈ {4, 6} for all i ∈ found(T ). If for all i, r i = 4, then by Lemma 18 we have that afrk(T ) ≥ 7. If r i = 6, for some i ∈ found(T ), then by Lemma 20 applied to T i , we obtain that r ij = 3 for some j ∈ found(T ). It turns out that r j = 6. Thus T ij has 18 blocks; so afrk(T ij ) ≥ 5. It follows that afrk(T ) ≥ 7, a contradiction. Hence, the claim follows.
Therefore, we assume that r k = 8 and so vol(T k ) = 10. Also afrk(T k ) = |found(T k )| = 5. For every i ∈ found(T ), r i is even (4, 6, or 8); hence, the volumes of T ki and T ki have the same parity. It follows that the number of elements with an odd replication in T k is the same as the number of elements with an odd replication in T k . However, the former is an odd number by Lemma 22(ii) and the latter is an even number by Lemma 22(iii), a contradiction. Now, we are ready to prove the main result of this section.
Theorem 24. If T is a simple [t]-trade with 1.5 · 2 t < vol(T ) < 2.5 · 2 t and vol(T ) = 2 t+1 , then the affine rank of T is at least t + 4.

Proof.
We proceed by induction on t. For t = 1, there is no trade satisfying the assumptions, and t = 2, 3 has been settled in Lemma 23. Hence we assume that t ≥ 4.
Since shifts do not change the affine rank of trades, we may assume that T is reduced. As T is reduced, r i ≤ vol(T )/2 < 2.5 · 2 t−1 for all i ∈ found(T ). If there exists some i ∈ found(T ) with r i = 2 t and r i > 1.5 · 2 t−1 , then T i is a simple [t − 1]-trade with vol(T i ) = 2 t and 1.5 · 2 t−1 < vol(T i ) < 2.5 · 2 t−1 . So by the induction hypothesis, afrk(T i ) ≥ t + 3. Therefore, afrk(T ) ≥ t + 4, and we are done. Hence we can assume that for all i ∈ found(T ), either r i = 2 t or r i ≤ 1.5 · 2 t−1 .
So it suffices to consider the following two cases.
Case 1. There exist some i ∈ found(T ) with r i = 2 t .
The following corollary will be used in the next section.

Proof. Suppose for a contradiction that T is a simple [t]-trade with vol(T )
By Theorem 24, afrk(T ) ≥ t + 4. On the other hand, let T ′ be the unitrade associated with T . By Lemma 19(ii), afrk(T ) = afrk(T ′ ) = t + 3, a contradiction.

· 2 t
Based on the characterization of codewords of Reed-Muller code with weights within the range 2 and 2.5 times the minimum distance by Kasami et al. [8], the following was obtained in [11].
Theorem 27. If the volume of a [t]-trade is between 2 · 2 t and 2.5 · 2 t , then it has one of the following forms: In Corollary 26, we showed that [t]-trades with volumes of the form (i) do not exist (except for i = t − 2 and t − 3 which can be represented in the form (ii) and (iii), respectively). In this section, we show by construction that they do exist with volumes of the forms (ii) and (iii). So the spectrum of volumes of [t]-trades in the range 2 · 2 t and 2.5 · 2 t is completely determined. For the construction, we employ the following observation of [11].
Lemma 28. Assume that (T + , T − ) and (T ′ + , T ′ − ) are two different simple [t]-trades such that Theorem 29. There exist simple [t]-trades of volumes: Proof. (i) Let Define T + 1 (T − 1 ) to be the set of vectors of T 1 with an odd (even) weight and T + 2 (T − 2 ) to be the set of vectors of T 2 with an even (odd) weight. We have is a [t]-trade similarly. For its volume we have as required.
From Corollary 26, Theorems 27 and 29, we have the following.
6 Characterization of small [t]-trades for t = 1, 2 We say that two trades are equivalent if one is obtained from the other by some permutation of the elements of V , some shifts, and, optionally, the swap of the two components T + , T − of the trade.
In this section we characterize [1]-trades of volume 3 and [2]-trades of volume 6 up to equivalence.

[1]-trades of volume 3
By the definition, a small [1]-trade has volume smaller than 4. Lemma 4 describes the [1]-trades of minimum nonzero volume 2; the remaining value is considered in the following simple theorem.
Proof. Let (T + , T − ) be a [1]-trade, then every element i occurs in the same number of blocks from T + and from T − . If this number is 2 or 3, then we consider the x i -shift, for which it is 1 or 0.
Making this for all elements, we get a [1]-trade satisfying the conditions from the conclusion of the theorem.

[2]-trades of volume 6
In the following four propositions, we define four types of [2]-trades of volume 6. The main result of this section states that every [2]-trade of volume 6 is of one of these four types.
Proposition 32. Assume that a [2]-trade T = (T + , T − ) of volume 6 is represented as can be empty and a relation of type Z i = Y j Y k is possible). Then, every extension T ′ of T has the same form, up to a shift.
Proof. We have By Lemma 7 and the definition, an extension (T ′ + , T ′ − ) has the form T ′ (Note that the multiset union ⊎ is essential here, as some blocks can have multiplicity 2; e.g., if XZ 1 = Y 2 Y 3 .) W.l.o.g., we may assume that vol(Q) ≤ 3 (otherwise, we consider the x s -shift). If it is 0, the statement holds trivially; 1 is not possible by Lemma 4. So it suffices to consider the following two cases.
, then every element of Y 1 occurs twice in the blocks of Q + . The same should be true for Q − ; so, either Q − contains XY 1 , or Q − = {Z i Z j , Z i Z k }. In the first case, utilizing the definition of a [1]-trade, we see that the second block of Q − is XY 1 Y 2 Y 3 , which is not a block from T − , a contradiction. In the second case, taking into account that Y 1 Y 2 Y 3 = Z i Z j Z k , we conclude that Y 1 = Z i , which does not fit the hypothesis of the proposition.
If Q + = {XZ 1 , XZ 2 } (similarly, {XZ 1 , XZ 3 } or {XZ 2 , XZ 3 }), then the elements of Z 3 do not occur in the blocks of Q + . The same should be true for Q − . So, Q − does not contain Z 1 Z 3 or Z 2 Z 3 . If it contains Z 1 Z 2 , then the second block is X, which is not from T − , again a contradiction. Therefore, Q + = {XY i , XY j } and w.l.o.g., Q + = {XY 1 , XY 2 }. But this leads to Z 1 Z 2 = Y 1 Y 2 , and from Z 1 Z 2 Z 3 = Y 1 Y 2 Y 3 we find that Z 3 = Y 3 , which contradicts the hypothesis of the proposition.
similarly, every remaining case), then we can assume that Q − = {XY i , Z j Z k } (the other cases are shown above). From We now see that every element occurs exactly twice in blocks of Q + ∪ Q − . By the definition of a [1]-trade, every element occurs exactly once in blocks of Q + (similarly, Q − ). But this means that Z 1 = Y 3 , a contradiction. Either Q + , or S + contains XZ i and XZ j for some different i and j. W.l.o.g. we can assume that Q + contains XZ 1 , XZ 2 . Consider the following two subcases.
(2a) Q + = {XZ 1 , XZ 2 , XZ 3 }. All elements of Z 1 Z 2 Z 3 occur exactly once in the blocks of Q + and, hence, in the blocks of Q − . So, Q − cannot have two blocks from Z 1 Z 2 , Z 1 Z 3 , Z 2 Z 3 and must have at least two blocks from XY 1 , XY 2 , XY 3 . The third block of Q − is uniquely determined and Q − = {XY 1 , XY 2 , XY 3 }. We see that the claim of the proposition holds with If j = 2, then W can only be XY 2 , in which case Then, the x s -shift of T has the required form.
If j = 3 and i = 3, then W = XZ 1 , which is not a block of T − .
Proposition 33. Assume that a [2]-trade T = (T + , T − ) of volume 6 is represented as where Y 1 , Y 2 , Y 3 are mutually disjoint nonempty sets, and likewise Z 1 , Z 2 , Z 3 are mutually disjoint nonempty sets, Y 1 , Y 2 , Y 3 , Z 1 , Z 2 , Z 3 are mutually different nonempty sets, and Y 1 Y 2 = Z 1 Z 2 . Then every extension T ′ of T has the same form, up to a shift.
Proof. We have Repeating the arguments of the previous proof, we conclude that we have to check all possibilities for a [1]-subtrade Q = (Q + , Q − ) of volume 2 or 3.
(the underlined blocks are from T − , the other are from T + ). We first note the following fact.
(*) The sets Q + and Q − have the same number of elements from each of X, Y , Z.
Indeed, since Y 3 and Z 3 are different, we have Y 3 \Z 3 = ∅ or Z 3 \Y 3 = ∅. Assume w.l.o.g. that Z 3 \Y 3 is not empty; i.e., it contains some element x i . By Lemma 7, Q + ∩ Z and Q − ∩ Z are the legs of a [0]-trade; hence, the cardinalities of this intersection are equal. Next, consider an element x j from Y 3 . If x j ∈ Z 3 , then, similar to the argument above, we obtain that |Q In any case, the whole statement of (*) follows. Assume that Q + has one block from X, say X, and one block from Y , say Y . Then, from (*), Q + also has one block from X, say X ′ , and one block from Y , say Y ′ . We have XX ′ = Z i Y j and Y Y ′ = Y k for some i, j, k ∈ {1, 2}. In any case, XX ′ Y Y ′ = Z l for some l ∈ {1, 2}, which contradicts Lemma 8. So, Q + cannot have one block from X and one from Y . Similarly, Q + cannot have one block from X and one from Z, or one block from Y and one from Z. The remaining possibilities satisfy the statement of the proposition: With these assumptions, X ′ , Y ′ , and Z ′ are uniquely determined by X, Y , and Z. It remains to consider the eight possibilities to choose X, Y , and The following two possibilities are in agree with the proposition statement: Consider the six other possibilities to choose X, Y , Z from X ∩ T + , Y ∩ T + , Z ∩ T + . For example, let Q + = {Z 2 , Y 2 Y 3 , Z 3 } (the other five cases are similar); so, where Y 1 , Y 2 , Y 3 , Z 1 , Z 2 are mutually disjoint nonempty sets. Then, every extension T ′ of T has the same form, up to a shift.
Proof. We have Repeating the arguments of the proofs of Propositions 32 and 33, we need to check all possibilities for a [1]-subtrade Q = (Q + , Q − ) of volume 2 or 3.
Similarly to the claim (*) in the proof of Proposition 33, we have (*) Q + and Q − have the same number of elements from each of Z and Z ′ . Now, assume that Q is a [1]-subtrade of volume 2 or 3. Consider the following four cases, which exhaust all possibilities.
Without loss of generality assume |Q + ∩Z| = 2. Necessarily we have |Q − ∩Z| = 2, and so Q + ⊇ is a [1]-trade, and we cannot add one more element to each leg keeping the [1]-trade property. So, vol(Q) = 2 and In this case we have The leg Q + has two intersecting blocks, but the blocks of Q − are mutually disjoint; we have an obvious contradiction with the definition of a [1]-trade.
Since ∅ ∈ T − , hence we reach at a contradiction.
Consider the following subcases. Lemma 8 we observe that the [1]-trade (Q + , Q − ) cannot have volume 2. So, Q + has two elements from Y , say Y i Y j and Y i Y k . By Lemma 8 we find Y i ∈ Q − , and so Proposition 35. Assume that , 2, 3} and X = ∅ Then, every extension of T has the same form, up to a shift.
Proposition 35 is a partial case of the following more general fact.
Proposition 36. Assume that whereσ is a [t − 1]-trade of volume less than 2 t (i.e., small) and X is a nonempty set, disjoint from the blocks ofσ (so, T is a small [t]-trade). Let T ′ be an extension of T . Then or whereσ ′ is an extension ofσ.
Proof. We have T ′ = x sκ + (T −κ), whereκ is a [t − 1]-subtrade of T . W.l.o.g., we can assume thatκ is small. Letκ p be the projection ofκ in X. Thenκ p is a small [t − 1]-trade, whose blocks are blocks ofσ. Let us prove the following claim: (*) Ifκ p is not void, then all blocks of the [t − 1]-tradeκ p +σ have even multiplicity.
Denote by a and b the number of different blocks ofσ of odd and even multiplicity, respectively. The volume ofσ is at least (a + 2b)/2; sinceσ is a small [t − 1]-trade, we have Denote by a ′ and b ′ the number of blocks ofκ p of odd multiplicity whose multiplicity inσ is odd and even, respectively. So, the number of blocks of odd multiplicity inκ p +σ is a − a ′ + b ′ .
Next, sinceκ p is a small non-void [t − 1]-trade, by Lemma 3 we have Now, using (12), (13), and the trivial fact that b ′ ≤ b, for the number a − a ′ + b ′ of oddmultiplicity blocks ofκ p +σ we have By Lemma 3, this number is 0. Hence (*) follows.
Ifκ p is void, we have (11). By (*), it remains to consider the case when all blocks of the [t − 1]-tradeκ p +σ have even multiplicity.
Theorem 37. Every [2]-trade of volume 5 or 6 have one of the forms described in Propositions 32-35.
In particular, Theorem 37 implies that there are no [2]-trades of volume 5, which is a known fact [6].
Proof. We proceed by induction on the number of the elements involved in the blocks of a trade. If this number is zero, then the statement is trivial (there are no non-void trades), which gives the induction base. Let us consider a [2]-trade T of volume 5 or 6. If it has a projection of volume 5 or 6, then by the inductive hypothesis the statement of the theorem holds for this projection. Hence, it is true for T , by Propositions 32-35.
If T has a void projection, then it has the form T = (1 − x i )σ, whereσ is a [1]-trade of volume 3. In this case, the statement is straightforward from Theorem 31.
It remains to consider the case when all projections have volume 4. For a given i, the i-projection has the form up to a shift. Then where α 000 , α 100 , α 010 , α 001 , α 110 , α 101 , α 011 , α 111 ∈ {1, x i } and V , W are some blocks with i ∈ V, W . The number of blocks of T with (or without) element i is at least 2 and at most 10; taking into account Lemma 7, it is 4, 6, or 8. So, the number p i of coefficients α ··· equal to x i is 2, 4, or 6. W.l.o.g. (up to the x i -shift) we may assume that it is p i = 2 or 4. The case of p i = 2, up to a shift and renaming X, Y , and Z, is exhausted by the Cases 1-3 below.
The last case is impossible because the four blocks x i , x i XY , x i Y Z, x i XZ have the same sign. We conclude that which has the j-projection of volume 6, contradicting our assumption.
(4b) The remaining subcase is |X| = |Y | = |Z| = 1. Each of V , W is one of 1, X, Y , Z, XY , XZ, Y Z, XY Z. It is not difficult to conclude that, up to a shift, which is the case of Proposition 32.

Computational results
In this section we present an algorithm to construct [t]-trades with a given foundation of size v. We implement this algorithm and enumerate all small [t]-trades for t ≤ 4.

Add
At the end, T will be the set of all s-small [t]-trades. Indeed, for every such trade T , consider the representation T = P + x v P ′ , where v ∈ found(P ), found(P ′ ). If P ′ is s-small, then T is added at Step 1.1 with T ′ = P + P ′ and T ′′ = P ′ . If P ′ is not s-small, then P is s-small, and T is added at Step 1.2 with T ′ = P + P ′ and T ′′ = P .
From T , we can choose a complete collection of nonequivalent s-small [t]-trades (to be exact, representatives of all equivalence classes). The graph isomorphism routine [15] is employed to deal with the equivalence rejection. See [9] for general technique of representing subsets of 2 V by graphs, for checking the equivalence. If we do not need the list of all trades, we can check equivalence at Steps 1.1 and 1.2, and collect only nonequivalent representatives. In this case, there is an obvious improvement: it is sufficient to consider either only nonequivalent [t]-trades T ′ , or only nonequivalent [t − 1]-trades T ′′ . However, the second component, T ′′ or T ′ , must be chosen from all different trades with corresponding parameters, and this approach does not allow to make all steps of the recursion by considering only nonequivalent representatives.

Proof of Lemma 22
For t = 2, we can further implement our algorithm to construct all [t]-trades T with 2 · 2 t ≤ vol(T ) ≤ 3 · 2 t and |found(T )| = 5. In particular, Lemma 22 is derived. The enumeration of these trades is given in the table below. vol.