SOME NEW TERNARY LINEAR CODES

Abstract: Let  q d k n , , code be a linear code of length n, dimension k and minimum Hamming distance d over GF (q). One of the most important problems in coding theory is to construct codes with best possible minimum distances. In this paper new one-generator quasi-cyclic (QC) codes over GF (3) are presented. Some of the results are received by construction X.


Introduction
Let an [n, k, d] q code be a linear code of length n, dimension k and minimum Hamming distance d over a finite field GF (q).One of the most important and fundamental problems in coding theory is to find the optimal values of the parameters of a linear code.
This optimization problem can be formulated in a couple of ways.For example, for fixed q, n and k we may wish to maximize the minimum distance d; or for given q, k and d to minimize the block length n.Let d q (n, k) denote the largest value of d for which there exists an [n, k, d] q code, and n q (k, d) be the smallest value of n for which there exists an [n, k, d] q code.Then an [n q (k, d), k, d] q code is called lengthoptimal and an [n, k, d q (n, k)] q code is called distance-optimal.Both length-optimal and distance-optimal codes are called optimal codes.The problem of finding the parameters of optimal codes is a very difficult one and has two aspectsone involves the construction of new codes with better minimum distances and the other is proving the nonexistence of codes with given parameters.It has been solved only over small finite fields for small dimensions and co-dimensions.
Computer search is often used in looking for codes with better minimum distances, but it is a well known fact that computing the minimum distance of a linear code is an NP-hard problem [15].Since it is not possible to carry out exhaustive searches for linear codes with large dimension, it is natural to focus one's effort on subclasses of linear codes, having rich mathematical structures.Quasi-cyclic (QC) codes are known to have such a structure and it has been shown in recent years that this subclass contains many new good linear codes ( [1,[4][5][6][7][8][9][10][12][13][14] and [E.Metodieva, N. Daskalova, Generating generalized necklaces and new quasi-cyclic codes, in preparation, 2017]).
Grassl [11] maintains a table with lower and upper bounds on minimum distances of linear codes over small finite fields GF(q) (q ≤ 9).When the constructed code has a minimum distance equal to the upper bound, it is optimal and there is no place for improvement in the table.When there is a gap between the minimum distance of the best-known code and the upper bound on the minimum distance, this is indicated in the table by listing both valuesd l and d u .Many of the best-known codes in these tables are QC codes.A code that attains a lower bound in the table is called a good code.A code that improves a lower bound in the table will be called a new code.
Another online table of linear codes is also maintained by Chen.Chen's table [3] contains only good and best-known quasi-cyclic and quasi-twisted codes (q ≤ 13).These two databases are updated when new codes are discovered.
The remainder of the paper is organized as follows.In Section 2, some basic definitions and facts on QC codes are presented.In Section 3, sixteen good one-generator QC codes (p ≥ 2) are constructed using an algebraic-combinatorial computer search.In Section 4 (Theorem 4.1), we use the codes presented in section 3, along with construction X, to obtain seventeen new ternary linear codes.In Theorem 4.2 five new codes are also presented.

Quasi-cyclic codes
A code C is said to be quasi-cyclic (QC or p-QC) if a cyclic shift of a codeword by p positions results in another codeword.A cyclic shift of an m-tuple (x 0 , x 1 , . . ., x m−1 ) is the m-tuple (x m−1 , x 0 , . . ., x m−2 ).The blocklength n of a p-QC code is a multiple of p, so that n = pm.
A matrix B of the form is called a circulant matrix.A class of QC codes can be constructed from m × m circulant matrices.In this case, the generator matrix G can be represented as where B i is a circulant matrix.
The algebra of m×m circulant matrices over GF (q) is isomorphic to the algebra of polynomials in the ring GF (q)[x]/(x m −1), with B being mapped to the polynomial, b formed from the entries in the first row of B. The b i (x)'s associated with a QC code are called the defining polynomials.
If the defining polynomials b i (x) contain a common factor which is also a factor of x m − 1, then the QC code is called degenerate.
The dimension k of the QC code is equal to the degree of h(x), where If the polynomial h(x) has degree m, the dimension of the code is m, and ( 2) is a generator matrix.
If deg(h(x)) = k < m, a generator matrix for the code can be constructed by deleting m − k rows of (2).
Let the defining polynomials of the code C have the following form where Then C is a degenerate, one-generator QC code having n = mp, and k = m − deg g(x) (see [14]).
In our constructions we will use the following well-known theorems.
when we have an [a, l, s] q code C 3 (by appending codewords from the latter code to cosets of the second code in the first code).
and append tails from a [a i , k i , δ i ] q code to the codewords of C , where the two tails of codewords correspond to the coset of , d 2 and d 0 , respectively, then there exists an

Good QC codes
In this section sixteen good one-generator QC codes (p ≤ 4) are constructed using a non-exhaustive algebraic-combinatorial computer search, similar to that in [1, 4-6, 8-10, 14].An important feature of these codes is that they have good subcodes and can be used for construction X.
In the following theorems the defining polynomials are listed with the lowest degree coefficient on the left.

The new codes
Let us look at an example, related to the next Theorem 4.1.We will show how the The generator matrix G 2 of the [140, 12, 75] 3 subcode C 2 has the same first row as the row given in Theorem 4.1.The generator matrix of the [11,5,6] 3 code is

G 2 0 * G 3 
[151, 17, 75] 3 code C is constructed.A generator matrix of a code C has the form G =   , where G 2 and G 3 are generator matrices of the codes C 2 and C 3 respectively, and ( * ) denotes l linearly independent codewords of a code C 1 .