On a class of repeated-root monomial-like abelian codes

In this paper we study polycyclic codes of length p1 × · · · × pn over Fpa generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in [10] on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables. 2010 MSC: 94B15,11T71,11T55


Introduction
Cyclic codes are said to be repeated-root when the codeword length and the characteristic of the alphabet are not coprime.Despite that it has been proved that in general they are asymptotically bad in some cases repeated-root cyclic codes are optimal and they have interesting properties.Massey et.al. have shown in [10] that cyclic codes of length p over a finite field of characteristic p are optimal.There also exist infinite families of repeated-root cyclic codes in even characteristic according to the results of [14].Also in [10] it has been pointed out that some repeated-root cyclic codes can be decoded using a very simple circuitry.Among other studies on repeated-root cyclic codes with several different settings are [1,2,7,8,11,12,14].

Monomial-like codes
Contrary to the simple-root case, there are repeated root cyclic codes of the form f (x) i where i > 1.Specifically, all cyclic codes of length p s over a finite field of characteristic p are generated by a single "monomial" of the form (x − 1) i , where 0 ≤ i ≤ p s (see [2,11]).In this paper, as a generalisation of these codes to several variables, we study cyclic codes of the form i.e.I (i1,...in) is the ideal of R generated by a single monomial of the form This paper is organised as follows.First we introduce some notation, give some definitions and prove some structural properties of the ambient space of a particular class of abelian codes in Section 2. In Section 3, we show thatmonomial like codes arise from product codes and we determine their Hamming distance.We describe their duals which yields a parity check matrix for these codes.In Section 4, we explain the relationship of the Hasse derivative with the dual of this type of codes.Finally in Section 5, we generalise the weight retaining property of monomials in single variable to the multivariable case.

The Ambient Space
Throughout the paper, we consider the finite ring as the ambient space of the codes to be studied unless otherwise stated.It is a well known fact that R is a local ring with maximal ideal (x 1 − 1, . . ., x n − 1).We define The elements of R can be identified uniquely with the polynomials of the form so throughout the paper, we identify the equivalence class with the polynomial f (x 1 , . . ., x n ).We shall consider a repeated-root code as just an ideal C of R. The length of the code is e. the number of nonzero coefficients of f (x 1 , . . ., x n ).The minimum Hamming distance of the code C is defined as

Monomial-like codes
In this paper we shall study a particular class of the codes over R called monomial-like codes given by an ideal generated by a single monomial of the form E. Martínez-Moro et al.
Note that not all the ideals in R can be generated by a single monomial of this form.
In one variable case, the minimum Hamming distance of C was computed in [11] and [2].It turns out that, in multivariate case, C (i1,...,in) can be considered as a product code of single variable codes.This decomposition allows us to express the minimum Hamming distance of C (i1,...,in) in terms of the Hamming distances of cyclic codes of length p sj .Definition 3.1.The product of two linear codes C, C over F p a is the linear code C ⊗C whose codewords are all the two dimensional arrays for which each row is a codeword in C and each column is a codeword in C .
The following are some well-known facts about the product codes. . Then are two generator matrices for C x and C y , respectively.
We identify the polynomial f (x, y) = 0≤i<n1,0≤j<n2 c ij x i y j ∈ F p a [x, y], with the codeword The elements of C = (x − 1) k1 (y − 1) k2 ⊂ R are exactly all the F p a -linear combinations of the elements of the set Now we consider G = G x ⊗ G y .Using the above identification for the rows of G, we obtain a basis for Monomial-like codes Corollary 3.3.Let r 1 , . . ., r n , i 1 , . . ., i n be positive integers and let Remark 3.4.
1. Note that the tensor product is associative in the sense that there is a natural isomorphism Thus we can remove all the parenthesis in Equation 6.
2. The reader can identify in Theorem and Corollary 3.3 as a polynomial version of the the fact that for The previous construction give us a straightforward result for the minimum distance of our codes as follows.
where d(C (ij ) ) is the minimum distance of the code (x Theorem 6.4] and [11,Theorem 1] in terms of p, a and i j .

Weight hierarchy of some two-variable cases
In some very special two-variable cases we can go slightly further and compute explicitly the whole weight hierarchy of the code.The r-th generalised Hamming weight d r (C) , 1 ≤ r ≤ k, of a F p -linear code C of dimension k is defined as the minimum of the cardinalities of the supports of all the subcodes (linear subspaces) of dimension r of C. We will define d 0 (C) = 0.The sequence {d r (C)} k r=0 is called the Hamming weight hierarchy of C.

Consider now C
2 −1) and let k 1 , k 2 the dimension as F p a -linear spaces of the codes C (i1) , C (i2) respectively.Using [13, Theorem 1] and since

Dual codes
Note that the elements of the form n k=1 (x k − 1) j k with j ∈ N n form a basis of F p a [x 1 , . . ., x n ] and the elements of this form with j k ≥ p s k for some k form a basis of ({x p s k − 1} n k=1 ).Let us consider ).This proves that the annihilator of C (i1,...,in) is ({(x − 1) p s k −i k } n k=1 ) and the dual of an ideal of R is exactly its annihilator.Therefore we have proved the following statement.Theorem 3.6.
Remark 3.7.Note that the above fact does not hold for arbitrary ideals of algebras of type and it relies on the fact that the n i = p si .
Let us construct an F p a -basis for C ⊥ .This will provide us a generator matrix for C ⊥ and hence a parity check matrix for C.

Now applying the inclusion-exclusion principle we obtain
Clearly the elements of B are F p a -linearly independent and |B| = |T |.On the other hand, we know, from Theorem 3.2, that dim(C (i1,...,in) ) = (p s1 −i 1 ) • • • (p sn −i n ).This implies that dim(C ⊥ (i1,...,in) ) = p s −dim(C (i1,...,in) ) which agree the cardinality of B, thus the set B is an F q -basis for C ⊥ .I.e., if we consider the vector representations of the elements of B, we obtain a generator matrix for C ⊥ and a parity check matrix for C.

Duality and the Hasse derivative
In this subsection we will show the natural relation between the Hasse derivative and the dual of monomial-like of codes.We begin by recalling the Hasse derivative which is used in the repeatedroot factor test.For a detailed treatment of the Hasse derivative, we refer to [4, Chapter 1] and [5,Chapter 5].Note that the standard derivative for polynomials over a field of positive characteristic, say p, is inappropriate because from the p th derivative on, the result is always zero.For this reason, it is more convenient to work with the Hasse derivative.Sometimes the Hasse derivative is also called the hyper derivative.Throughout this section, we will use the convention that a b = 0 whenever b > a.
As an immediate consequence, we have the following theorem.
For g(x 1 , . . ., x n ) ∈ R, let u g be the vector representation of the polynomial with respect to the fixed ordering.Then the dot product of w a and u g gives us the evaluation of the Hasse derivative of g(x 1 , . . ., x n ) at (1, . . ., 1) in the direction a, i.e., w a • u g = D [ a] (g)(1, . . ., 1).If we construct the matrix H whose rows are the vectors w a where a ∈ Q and Q is as in Theorem 4.3 then H is an alternative parity check matrix for the code C (i1,...in) by Theorem 4.3.

A generalisation of the weight retaining property
In [10], the so-called weight retaining property of polynomials over finite fields was stated and proved.This property turned out to be very useful for determining the Hamming distance of cyclic codes.
In this section, we give a generalisation of the weight retaining property to multivariate polynomials.We prove that the Hamming weight of any F p a -linear combination of the monomials (x 1 − c 1 ) i1 • • • (x n − c n ) in is greater than or equal to the Hamming weight of the "minimal" nonzero term, where a "minimal" term is the one that is not divisible by the rest of the nonzero terms of the summation.
First, we consider the case in one variable which was studied in [10].The weight retaining property of (x − c) i is given in the following two theorems.It is not hard to see that Theorem 5.1 is equivalent to the following theorem.

2 .
If G and G are generator matrices of C and C respectively, then G ⊗ G is a generator matrix of C ⊗ C , where ⊗ denotes the Kronecker product of matrices and the codewords of C ⊗ C are seen as concatenations of the rows in arrays in C ⊗ C .Theorem 3.2.Let n 1 , n 2 be positive integers and let was shown in [10, Theorem 5] that C (i1) is a Maximum Distance Separable (MDS) code.The weight hierarchy of a MDS code C is completely determined by its length n and dimension k as d r (C) = n − k + r for r = 1, 2, . . ., k, see for example [6, Theorem 7.10.7].

Theorem 5 . 1 .
[10,  Theorem 1.1 and Theorem 6.1] Let L be any nonempty finite subset of non-negative integers with least integer i min and letf (x) = i∈L b i (x − c) iwhere c and each b i are nonzero elements of F p a .Then w(f (x)) ≥ w((x − c) imin ).