Construction of quasi-twisted codes and enumeration of defining polynomials

Let dq(n, k) be the maximum possible minimum Hamming distance of a linear [n, k] code over Fq. Tables of best known linear codes exist for small fields and some results are known for larger fields. Quasi-twisted codes are constructed using m×m twistulant matrices and many of these are the best known codes. In this paper, the number of m × m twistulant matrices over Fq is enumerated and linear codes over F17 and F19 are constructed for k up to 5. 2010 MSC: 94B05, 94B65, 94A55


Introduction
Let F q denote the finite field of q elements, and V (n, q) the vector space of n-tuples over F q . A linear [n, k] code C of length n and dimension k over F q is a k-dimensional subspace of V (n, q). The elements of C are called codewords. The (Hamming) weight of a codeword is the number of non-zero coordinates, and the minimum distance of C is the smallest weight among all non-zero codewords of C. An [n, k, d] code is an [n, k] code with minimum distance d. Let A i be the number of codewords of weight i in C. Then the numbers A 0 , A 1 , . . . , A n are called the weight distribution of C.
A central problem in coding theory is that of optimizing one of the parameters n, k and d for given values of the other two. One can find d q (n, k), the largest value of d for which there exists an [n, k, d] code over F q , or n q (k, d), the smallest value of n for which there exists an [n, k, d] code over F q . A code which achieves either of these values is called optimal. Tables of best known linear codes exist for q = 2 to 9 [6], 11 [3] and 13 [4].
The Griesmer bound is a well-known lower bound on n q (k, d) where x denotes the smallest integer ≥ x. For k ≤ 2, there exist codes that attain equality in the Griesmer bound for all q and d. The Singleton bound [12] is a lower bound on n q (k, d) and is given by Codes that meet this bound are called maximum distance separable (MDS). MDS codes exist for all values of n ≤ q + 1. Thus, for q = 17 MDS codes exist for all lengths 18 or less, and for q = 19 all lengths 20 or less. Note that all MDS codes are optimal. For larger lengths and dimensions, far less is known about codes over F 17 and F 19 .
A linear code C is said to be quasi-twisted (QT) if a constacyclic shift of any codeword by p positions is also a codeword in C [8]. Note that quasi-twisted codes generalize the classes of constacyclic codes (p = 1), quasi-cyclic codes (λ = 1), cyclic codes (λ = 1, p = 1), and negacyclic codes (λ = −1, p = 1). The length of a QT code considered here is n = mp. With a suitable permutation of coordinates, many QT codes can be characterized in terms of m × m twistulant matrices. In this case, a QT code can be transformed into an equivalent code with generator matrix where B i , i = 0, 1, . . . , p − 1, is an m × m twistulant matrix (also known as a constacyclic matrix), over F q of the form [9] where λ ∈ F q \{0} and b i , 0 ≤ i ≤ m − 1, are elements of F q . When λ = 1, a twistulant matrix is a circulant matrix, and when λ = −1, a twistulant matrix is known as a negacirculant matrix [8].
The algebra of m × m twistulant matrices over F q is isomorphic to the algebra of polynomials in the ring formed from the entries in the first row of B i . The b i (x) associated with a QT code are called the defining polynomials [7]. The set {b 0 (x), b 1 (x), . . . , b p−1 (x)} defines an [mp, m] QT code with k ≤ m.

Defining polynomials
The construction of QT codes requires a representative set of defining polynomials. These are the equivalence class representatives of a partition of the set of polynomials of degree less than m. For defining polynomials, multiplication by a non-zero element of F q does not change the weight and hence does not change the equivalence class. Thus, two polynomials r j (x) and r i (x) are said to be equivalent if for some integer l ≥ 0 and scalar γ ∈ F q \{0}.
A closed-form expression for the number of defining polynomials is now given. Let g be the permutation (1, 2, . . . , m) so that g maps i to i + 1, for 1 ≤ i ≤ m − 1, and m to 1. Therefore, g i , 1 ≤ i ≤ m, is also a permutation and has order m gcd(m,i) in the symmetric group of degree m. Thus, the action of g on the m-tuple x = (x 1 , x 2 , . . . , x m ) changes x to (x m , x 1 , x 2 , . . . , x m−1 ) where x i ∈ F q . Now let λg be such that the action of λg on the m-tuple changes x to (λx m , x 1 , x 2 , . . . , x m−1 ), the action of (λg) 2 on x results in (λx m−1 , λx n , x 1 , x 2 , . . . , x m−2 ), and similarly for other powers. Then the order of λg is Ord(λ)m, where Ord(λ) is the order of λ in F q . Further, let t(λg), t ∈ F q \ {0}, be such that it changes x to (tλx m , tx 1 , tx 2 , . . . , tx m−1 ). The action of t(λg) 2 changes x to (tλx m−1 , tλx m , tx 1 , tx 2 , . . . , tx m−2 ), and similarly for other powers. The equivalence relation is induced by the action of the group consisting of the elements t(λg) i , 1 ≤ i ≤ Ord(λ)m, t ∈ F q \ {0}. Distinct equivalence classes correspond to distinct orbits under the action of this group and so can be enumerated using Burnside's Lemma [5,9].
Theorem 3.2. The number of words of length m over the alphabet F q fixed by t(λg) i for some fixed Proof. Let x = (x 1 , x 2 , . . . , x m ) be a word of length m over F q . Then the relation between the components of x before and after the action of t(λg) i is . .
, the relation between the components of x is . .
Thus in general, the relation between the components of x is . .
Let gcd(m, i) = h. Then, from the expressions above, the orbit of x m is and so we also have Similar expressions exist for x m−2 , x m−3 , . . . , x m−h+2 , and in general Proof. There are (q − 1)Ord(λ)m permutations given by t(λg) i . Thus, by Burnside's lemma [5], the number of orbits of words of length m over an alphabet of size q is equal to the average number of words fixed by each t(λg) i , 1 ≤ i ≤ Ord(λ)m, t ∈ F q \ {0}. Therefore, we have where |Fix t(λg) i | denotes the number of words fixed by t(λg) i .
From Theorem 1, the number of words fixed by t(λg) i is either q gcd(m,i) or 1 depending on whether t m gcd(m,i) λ i gcd(m,i) = 1 or not. Therefore  Note that setting λ = 1 in (4) gives the number of defining polynomials for quasi-cyclic codes [14] M q,1 (m) = 1 Table 1 gives the number of defining polynomials over F 2 , F 3 , F 5 , F 7 , and F 11 with λ = 1, Table  2 gives the number of defining polynomials over F 4 , F 8 , F 9 , and F 16 with λ = 1, and Table 3 gives the number of defining over F 13 , F 17 , and F 19 with λ = 1. To illustrate the effect of λ, Tables 4 and 5 give the number of defining polynomials over F 3 and F 4 .

Quasi-twisted codes over F 17 and F 19
In this section, the defining polynomials given above are used to construct quasi-twisted codes over F 17 and F 19 . The number of defining polynomials over F 17 for m = 1 to 5 is given in Table 6 and over F 19 for m = 1 to 5 in Table 7. Note that the zero polynomial is not considered in constructing codes.
Considering a code structure (i.e. QT), results in a search space that is smaller than for the general code design problem. The more restrictions on the structure, the smaller the search, but this creates a tradeoff since good codes may be missed if too much structure is imposed on the code. The QT codes presented here were constructed using a stochastic optimization algorithm, namely tabu search, which is similar to that in [10,11,14]. By restricting the search for good codes to the class of QT codes, and using a stochastic heuristic, codes with high minimum distance can be found with a reasonable amount of computational effort. Based on the results obtained here, this approach provides a good tradeoff.

Conclusion
Closed-form expressions for the number of twistulant matrices and corresponding defining polynomials were given. These polynomials were used in the construction of quasi-twisted codes, and several new optimal codes were obtained.