We present the Linear Codes Weight Invariant Library (LinCodeWeightInv) for optimized computing of the weight distribution and other weight characteristics of a random linear code (minimum distance, number of codewords with a given weight, etc.). The presented library is developed for linear codes over finite fields with up to 64 elements. We use two main methods for optimizations—efficient algorithms for generating the codewords and integration of extended vector registers with SSE4.1, AVX2, and AVX512 instruction sets for x86 architectures and NEON instructions set for ARM. The LinCodeWeightInv is compared to other software systems for linear codes over finite fields. Comparing our library to the Magma software, we get between 1.3 and 4 times faster execution times for \(\mathbb{F}_{2}\) and \(\mathbb{F}_{5}\) and up to 49 depending on the field and the code length. Comparing the LinCodeWeightInv library to the open source software GAP, we observe a reduction in computation time by factors between 5 and 7 for \(\mathbb{F}_{2}\) . For other finite fields, we observe more than 100 times faster execution times.

Two algorithms for the classification of linear codes over finite fields are presented. One of the algorithms is based on canonical augmentation and the other one on lattice point enumeration. New classification results over fields with 2, 3 and 4 elements are obtained.

An efficient isomorph-free generation algorithm for classification of binary self-dual codes with the minimum distance d = 4 is presented. It is combined with another isomorph-free generation algorithm for classification of self-dual codes with the minimum distance d ≥ 6. A complete classification of all binary self-dual codes of length 40 is given as a result.

An efficient algorithm for classification of binary self-dual codes is presented. As an application, a complete classification of the self-dual codes of length 38 is given.

All linear binary codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact bounds for binary linear codes. Primarily two algorithms considering the dual codes are used, namely extension of dual codes with a proper coordinate, and a fast algorithm for finding a maximum clique in a graph, which is modified to find a maximum set of vectors with the right dependency structure.

The aim of this work is a systematic investigation of the possible parameters of quasi-perfect (QP) binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes. First, we give a list of infinite families of QP codes which includes all binary, ternary, and quaternary codes known to us. We continue further with a list of sporadic examples of binary and ternary QP codes. Later we present the results of our investigation where binary QP codes of dimensions up to 14 and ternary QP codes of dimensions up to 13 are classified.

Several methods for classifying self-orthogonal codes up to equivalence are presented. These methods are used to classify self-orthogonal codes with largest possible minimum distance over the fields $\mathbb{F}_3$ and $\mathbb{F}_4$ for lengths $n \leq 29$ and small dimensions (up to 6). Some properties of the classified codes are also presented. In particular, an extensive collection of quantum error-correcting codes is obtained.

We prove that a linear binary code with parameters [34,24,5] does not exist. Also, we characterize some codes with minimum distance 5 and 6.

Let d/sub 3/(n,k) be the maximum possible minimum Hamming distance of a ternary [n,k,d]-code for given values of n and k. We describe a package for code extension and use this to prove some new exact values of d/sub 3/(n,k). Moreover, we classify the ternary [n,k,d/sub 3/(n,k)]-codes for some values of n and k.

We prove the nonexistence of binary [69,9,32] codes and construct codes with parameters [76,9,34],[297,9,146], and [300,9,148]. These results show that n(9,32)=70, n(9,34)/spl les/76,n(9,146)=297, and n(9,148)=300, where n(k,d) denotes the smallest value of n for which there exists an [n,k,d] binary code. We also present some codes of minimum distance 32 and some related codes.

Let n(8,d) be the smallest integer n for which a binary linear code of length n, dimension 8, and minimum distance d exists. We prove that n(8,18)=42, n(8,26)=58, n(8,28)=61, n(8,30)=65, n(8,34)=74, n(8,36)=77, n(8,38)=81, n(8,42)=89, and n(8,60)=124. After these results, all values of n(8,d) are known.

We construct extremal self-dual codes with lengths 44, 50, 54, and 58. They have weight enumerators for which extremal codes were previously not known to exist. Two methods are used for constructing the codes using self dual codes of same or smaller length. To obtain the codes we use a combinatorial optimization search.

A method to design binary self-dual codes with an automorphism of order two without fixed points is presented. Extremal self-dual codes with lengths 40, 42, 44, 54, 58, 68 are constructed. Many of them have weight enumerators for which extremal codes were previously not known to exist.

Constructions of [162,8,80] and [159,8,78] codes are given. This solves the open problems of finding the minimum length of binary codes of dimension S and minimum distances 78 and 80, respectively.

Let n/sub q/(k,d) be the smallest integer n for which there exists a linear code of length n, dimension k, and minimum distance d, over a field of q elements. In this correspondence we determine n/sub 5/(4,d) for all but 22 values of d.

Let d/sub q/(n,k) be the maximum possible minimum Hamming distance of a q-ary [n,k,d]-code for given values of n and k. It is proved that d/sub 4/ (33,5)=22, d/sub 4/(49,5)=34, d/sub 4/(131,5)=96, d/sub 4/(142,5)=104, d/sub 4/(147,5)=108, d/sub 4/(152,5)=112, d/sub 4/(158,5)=116, d/sub 4/(176,5)/spl ges/129, d/sub 4/(180,5)/spl ges/132, d/sub 4/(190,5)/spl ges/140, d/sub 4/(195,5)=144, d/sub 4/(200,5)=148, d/sub 4/(205,5)=152, d/sub 4/(216,5)=160, d/sub 4/(227,5)=168, d/sub 4/(232,5)=172, d/sub 4/(237,5)=176, d/sub 4/(240,5)=178, d/sub 4/(242,5)=180, and d/sub 4/(247,5)=184. A survey of the results of recent work on bounds for quaternary linear codes in dimensions four and five is made and a table with lower and upper bounds for d/sub 4/(n,5) is presented.