A database of linear codes over F 13 with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm

: Error control codes have been widely used in data communications and storage systems. One central problem in coding theory is to optimize the parameters of a linear code and construct codes with best possible parameters. There are tables of best-known linear codes over ﬁnite ﬁelds of sizes up to 9. Recently, there has been a growing interest in codes over F 13 and other ﬁelds of size greater than 9. The main purpose of this work is to present a database of best-known linear codes over the ﬁeld F 13 together with upper bounds on the minimum distances. To ﬁnd good linear codes to establish lower bounds on minimum distances, an iterative heuristic computer search algorithm is employed to construct quasi-twisted (QT) codes over the ﬁeld F 13 with high minimum distances. A large number of new linear codes have been found, improving previously best-known results. Tables of [ pm, m ] QT codes over F 13 with best-known minimum distances as well as a table of lower and upper bounds on the minimum distances for linear codes of length up to 150 and dimension up to 6 are presented.


Introduction and motivation
Let [n, k, d] q denote a linear code of length n, dimension k and minimum distance (weight) d over the finite field F q .A central and fundamental problem in coding theory is to find the optimal values of the parameters of a linear code and construct codes with these parameters.The problem can be formulated in a few different ways.For example, we may wish to maximize the minimum distance d for the given block length n and dimension k; or minimize the block length n for the given dimension k and minimum distance.Let d q (n, k) denote the largest value of d for which there exists an [n, k, d] code over F q , and n q (k, d) the smallest value of n for which there exists an [n, k, d] code over F q .An [n, k, d] code is called optimal (or length-optimal) if its block length n equals n q (k, d), or if its minimum distance d equals d q (n, k) (also called distance-optimal).This optimization problem is very difficult.In general, it is only solved for the cases where either k or n − k is small.Computers are often used in searching for codes with best parameters but there is an inherent difficulty: computing the minimum distance of a linear code is computationally intractable [19].Since it is not possible to conduct exhaustive searches for linear codes if the dimension is large, researchers often focus on promising subclasses of linear codes with rich mathematical structures.As a generalization to cyclic and consta-cyclic codes, quasi-cyclic (QC) and quasi-twisted (QT) codes are known to have this characteristic.They have been shown to contain many good linear codes.With the help of modern computers, many record-breaking QC and QT codes have been constructed [2]- [13].However, the problem still becomes intractable as the dimension and the block length of the code get large.The records of best-known linear codes are available.For example, the online database [21] is one that is commonly referred to.It contains records of best-known codes over F q for q ≤ 9 together with upper bounds on d q (n, k).The Magma software [20] also contains a similar database.The online database of QT codes contains best-known QC and QT codes [24].These databases are updated as new codes are discovered.
Another reason for the interest in codes over F 13 is the connection between linear codes and finite geometries.Codes of dimension 3 are closely related to arcs in a projective geometry, and a lot of research has been carried out on projective codes of dimension 3 over finite fields of size up to 19 [4].
Finally, Venkaiah and Gulliver [13] used the tabu search to construct quasi-cyclic codes over F 13 of dimensions up to 6 and lengths less than 150.They constructed many QC codes of the form [pk, k], for over F 13 , and presented their results in several tables (one for each value of k).These tables constitute the most comprehensive set of best-known linear codes over F 13 to date.
In this paper, we present a database of linear codes over F 13 for lengths ≤ 150 and dimensions 3 ≤ k ≤ 6.We employed an iterative, heuristic algorithm [15] to conduct a computer search to produce new codes.With this algorithm, a large number of new QC and QT codes have been constructed many of which improve the previous results.We achieve improvements on the parameters of the codes presented in [13] in many cases.Combining the results presented in [13] with the new codes we have found, we create a comprehensive database of best-known linear codes over F 13 .To the best of our knowledge, this is the first time such a database appears in the literature.
The remainder of the paper is organized as follows.In Section 2, some basic definitions and facts on QT codes are presented.In Section 3, the iterative heuristic algorithm that is used to find good QT codes is described.Next, a database of linear codes over F 13 with minimum distance bounds is presented.The paper contains several tables: tables of new, improved QC and QT codes, maximum known minimum distances for QT [pm, m] codes, optimal QT codes, as well as a comprehensive table of lower and upper bounds on linear codes over F 13 that covers the range n ≤ 150 and 3 ≤ k ≤ 6.With these concrete results, this work can serve as a foundation for future research on linear codes over F 13 (e.g. a more comprehensive database).

Quasi-twisted codes
A linear q-ary [n, k, d] code is said to be α-consta-cyclic if there is a non-zero element α of F q such that for any codeword (a 0 , a 1 , . . ., a n−1 ), a consta-cyclic shift by one position, that is (αa n−1 , a 0 , . . ., a n−2 ), is also a codeword [14].Therefore, consta-cyclic codes are a generalization of cyclic codes, or a cyclic code is an α-consta-cyclic code with α = 1.A consta-cyclic code can be defined by a single generator polynomial.A code is said to be quasi-twisted (QT) if a consta-cyclic shift of any codeword by p positions is still a codeword.Thus a consta-cyclic code is a QT code with p = 1, and a quasi-cyclic (QC) code is a QT code with α = 1.The length n of a QT code is a multiple of p, i.e., n = pm for some positive integer m.
An α-consta-cyclic matrix of order n, also called a twistulant matrix, is defined as Twistulant matrices are basic components in the generator matrix for a QT code.The algebra of n × n consta-cyclic matrices over F q is isomorphic to the algebra of the quotient ring F q [x]/(x n − α) if C is mapped onto the polynomial formed by the elements of its first row, c(x with the least significant coefficient on the left.The polynomial c(x) is also called the defining polynomial of the matrix C. A twistulant matrix is called a circulant matrix if α = 1.
The generator matrix of a QT code can be transformed into rows of twistulant matrices by a suitable permutation of columns.Most research has been focused on 1-generator and 2-generator QT codes.The generator matrices for 1-generator and 2-generator QT codes consist of one row of twistulant matrices and two rows of twistulant matrices, respectively, where G j and G ij are twistulant matrices, for j = 0, 1, 2, . . ., p1 and i = 1, 2. Let g i,j (x) and g i,j (x) be the defining polynomials for the corresponding twistulant matrices G j and G ij .Then, the defining polynomials (g 0 (x), where k, the dimension of the code, is the rank of the generator matrix G.In Magma algebra system [20], the number of generators is called the height.The parameters of all the codes presented in this paper have been verified by Magma.

The search algorithm and new QT codes over F 13
As a generalization to cyclic codes and consta-cyclic codes, quasi-cyclic (QC) codes and quasi-twisted (QT) codes have been known to contain many good codes.In fact, many record-breaking linear codes have been obtained from these classes [2]- [13].
Gulliver et al. [4,5,9,13] have done much work on the computer searches for good QC and QT codes.By eliminating the equivalent generator polynomials, and eliminating all redundant information polynomials, an r × s weight matrix W is used in the constructions, as given below, where c k (x) is the kth generator polynomial, i j (x) is the jth information polynomial, w jk is the Hamming weight of i j (x)c k (x) , m is the size of the twistulant matrix and α is the shift constant.To construct a good QT [pm, k] code, their algorithm selects a set of p columns among s columns such that the set of columns maximizes the smallest row sum of the corresponding p columns.When p and s are large, it is not possible to examine all (s, p) combinations.Gulliver's search is initialized with an arbitrary [pm, k] code (usually a good one) with p columns (or generator polynomials).To improve the code, a new column is found to replace one presently in the code so that the minimum distance is increased.Later on, a stochastic optimization called tabu search has been used to construct good QC or QT codes by Gulliver and Östergård [9], and Daskalov et al. [10].In a recent paper, Venkaiah and Gulliver [13] used the tabu search to find good QC codes over F 13 .
On the other hand, a method to obtain a weight matrix from a consta-cyclic simplex code of composite length was recently presented in [15].The resulting weight matrix is cyclic, and therefore only one row is required to be in the memory during the search.A new iterative heuristic search is also presented, and many good QT codes have been constructed [15].In this work, the algorithm from [15] is applied to both the weight matrix defined by Gulliver's method and the weight matrix derived from the consta-cyclic simplex code as given in [15].As a result, many good QT codes have been obtained, allowing us to establish a database of linear codes over F 13 with the range of parameters described above.
Given an r × s weight matrix W = (w ij ).The iterative algorithm tries to find a sequence of good QT [im, k] q codes, i = 1, 2, . . ., t, where t < s.The basic idea of the algorithm is to extend a QT [(i − 1)m, k] q code by one more column to obtain a good QT [im, k] q code, for i = 2, 3, . . ., t.The algorithm is executed for a specified number of iterations.The algorithm records the best codes found so far, and stores them in files.When the algorithm stops, a summary of the codes found is presented.In the execution of the algorithm, the selection of columns is important as it determines if good codes can be found quickly.In order to avoid exhaustive search, we use a heuristic method to implement the selection.At each iteration, to obtain the best possible minimum distance for a QT [im, k] q code, we select a column that results in the largest minimum row sum (it is also the minimum distance of the constructed code).If there is more than one column that gives the same best minimum distance, we count how many such rows that result in the minimum row sum.We choose the column that will have the smallest number of such rows, since it is expected that such a selection will provide a better chance to get a good QT [(i + 1)m, k] q code in the next extension.In this way, the algorithm is greedy and heuristic.If there is more than one choice, a column is selected at random among suitable choices.So the algorithm contains some randomization.
The effectiveness of this iterative heuristic search algorithm is evident from the fact that a large number of new QT codes over F 13 for k = 3, 4, 5, and 6 have been obtained as a result of the application of the algorithm.The new codes improve the previously known results.
Table 1 lists the new QT codes over F 13 that have larger minimum distances than the corresponding codes given in [13].The defining polynomials are listed with the lowest degree coefficient on the left, and the finite field F 13 elements 10, 11, 12 are denoted by A, B and C (as commonly used in a hexa-decimal system).For example, C024A9 corresponds to the polynomial 12 + 2x 2 + 4x 3 + 10x 4 + 9x 5 .
Table 2 summarizes the maximum known minimum distances for QT [pm, m] codes over F 13 for p up to 25.The authors can provide all best known QT codes for n up to 255, upon request.Most entries in the table are from the results in [13], and the entries labeled with superscript "e" are new codes found with the algorithm in this paper.All codes with k = 6 are constructed from the weight matrix derived from the consta-cyclic simplex [402234, 6, 371293] code.Since the weight matrix is cyclic, only one row of 402234/6 = 67039 elements is required to be stored in memory.This makes it easier to search for good QT codes with k = 6 (otherwise, the weight matrix is too big to fit in the memory).
4. A database of linear codes over F 13 with minimum distance bounds 4.1.

Lower bounds on minimum distance
Since there are no good, general analytical lower bounds available for the parameters of a linear code, the lower bounds on minimum distances have been established by explicitly constructing the codes [1].As commented earlier, constructing good linear codes is a difficult task because finding the minimum distance of a linear code is computationally expensive [19].Therefore, researchers focus on certain promising classes of codes with rich mathematical structure.The class of QT codes has been an excellent source for producing best-known codes [2]- [13].Constacyclic codes are a special case of QT codes.Following the approach given in [22], we have been able to compute all constacyclic codes exhaustively for most lengths since the dimension is restricted to 3 ≤ k ≤ 6.Some of the best-known (or optimal) codes are constacyclic.
Another tool that can be used to obtain more new codes from existing codes in a computationally efficient way is to apply standard construction methods to derive codes from known codes, such as puncturing, shortening and extending [1].With the codes constructed in [13], the new QT codes over F 13 presented in the previous section, as well as the standard construction methods to derive new codes from existing codes, we are able to create a comprehensive table of lower bounds on the minimum distances for linear codes over F 13 with dimensions between 3 and 6 and block length n up to 255.Table 3

includes the lower bounds for block lengths up to 150.
There is a connection between best-known linear codes and projective geometry.An (n, r)-arc in P G(k − 1, q) is a set of n points K with the property that every hyperplane is incident with at most r points of K and there is some hyperplane incident with exactly r points of K.It is known that there exists a projective [n, 3, d] q code if and only if there exists an (n, n − d)-arc in P G(2, q) [13].Ball [17] maintains an online table of bounds on the sizes of (n, r)-arcs in P G(2, q) for q ≤ 19 .From that table, one can obtain lower bounds on the minimum distances of linear codes of dimension 3. Some of the entries in Table 3 for k = 3 can be derived from [17].Table 4 lists the defining polynomials for the new codes found in this paper and that are used to establish the lower bounds in Table 3.There are 7 new 2-generator QT codes with k = 6 and m = 3 that are used to derive the lower bounds in Table 3.

Upper bounds on minimum distance
We also determined upper bounds on the minimum distances by applying the standard bounds (such as Griesmer, Elias, Sphere Packing etc.) [1] and taking the best result for each parameter set.In the range of parameters considered here, Griesmer bound turned out to be the best for most of the cases except that in some cases the Levenshtein bound performed better.When a code whose minimum distance equals to the upper bound, an optimal code is constructed and there is no room for improvement in the table.When there is a gap between the minimum distance of a best-known code and the upper bound on the minimum distance, this is indicated in the table by listing the both values.For example, for a [51,4]-code, the minimum distance of a best-known code is 43 whereas the theoretical upper bound is 45.It is worth noting that the theoretical upper bound may be unattainable.To save the space, only entries for the block length n up to 150 are given below (Table 3).Interested readers can obtain the full table from the authors.

Some optimal codes over F 13
Table 3 presents the lower and upper bounds on d 13 (n, k) for k up to 6.Many bounds are attained.It is possible that some of the current upper bounds may be improved and more codes may turn out to be optimal.In the rest of this section, we give more details on the optimal codes in Table 3.
With the algorithm given in the last section, many QC codes with k = 3 have been constructed whose minimum distances meet the Griesmer bounds, and thus are optimal.Table 5 lists those optimal QC [pm, 3] codes that do not appear in [13].It should be noted that codes with these parameters were not constructed in the QC form [17,23]. Codes constructed in QC or QT form have advantages in practical implementation.Table 6 lists optimal QT [pm, k] codes for k = 4, 5 and 6, over F 13 , and their defining polynomials.With the upper bounds given in Table 3, we now know that the QC [20,4,16] and [28,4,23] codes constructed in [13] are optimal, since they reach the upper bounds.The optimal [153, 4, 139] code is included here, since two other optimal codes are obtained from it by puncturing: [150, 4,136] and [149,4,135] codes.The optimal [15,6,9] code given in the table is a 2-generator QT code with shift constant 6, and is constructed with the method given in [15].With these codes, and results on (n, r)-arcs, the exact values on d 13 (n, k) in Table 3 are established.

Conclusion
In this paper, we present the construction of a large number of new QT codes over F 13 obtained by an iterative heuristic search algorithm recently introduced.The results are presented in several tables.Combining the new results with earlier work on linear codes over F 13 , a database of linear codes over F 13 with both lower and upper bounds on the minimum distances is presented for the first time.We hope that the results presented in this paper serve as a basis for future study on codes over F 13 .

Table 3
Lower and upper bounds on minimum distances for linear codes over F13