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In this article, we present a new construction of evaluation codes in the Hamming metric, which we call twisted Reed–Solomon codes. Whereas Reed–Solomon (RS) codes are MDS codes, this need not be the case for twisted RS codes. Nonetheless, we show that our construction yields several families of MDS codes. Further, for a large subclass of (MDS) twisted RS codes, we show that the new codes are not generalized RS codes. To achieve this, we use properties of Schur squares of codes as well as an explicit description of the dual of a large subclass of our codes. We conclude the paper with a description of a decoder, that performs very well in practice as shown by extensive simulation results. [ABSTRACT FROM AUTHOR]

A q -ary code of length n , size M , and minimum distance d code. An (n,q^{k},n-k+1)_{q} code is called a maximum distance separable (MDS) code. In this paper, some MDS codes over small alphabets are classified. It is shown that every (k+d-1,q^{k},d)_{q} code with k\geq 3 , d \geq 3 , is equivalent to a linear code with the same parameters. This implies that the (6,5^4,3)5 code and the (n,7^{n-2},3)7 MDS codes for n\in \6,7,8\ are unique. The classification of one-error-correcting 8-ary MDS codes is also finished; there are 14, 8, 4, and 4 equivalence classes of (n,8^n-2,3)8 codes for $n=6,7,8$ , and 9, respectively. One of the equivalence classes of perfect (9,8^{7},3)_{8} codes corresponds to the Hamming code and the other three are nonlinear codes for which there exists no previously known construction. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

A binary code C ⊆ 픽2n with minimum distance at least d and codewords of Hamming weight w is called an (n, d, w) constant weight code. The maximum size of an (n, d, w) constant weight code is denoted by A(n, d, w), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n, d, w) constant weight codes are here classified up to equivalence for = 4, n ≤ 12; d = 6, n ≤ 14; d = 8, n ≤ 17; d = 10, n ≤ 20 (with one exception); d = 12, n ≤ 23; d = 14, n ≤ 26; d = 16, n ≤ 28; and d = 18, n ≤ 28. Moreover, several new upper bounds on A(n, d, w) are obtained, leading among other things to the exact values A(12, 4, 5) = 80, A(15, 6, 7) = 69, A(18, 8, 7) = 33, A(19, 8, 7) = 52, A(19, 8, 8) = 78, and A(20, 8, 8) = 130. Since A(15, 6, 6) = 70, this gives the first known example of parameters for which A(n, d, w - 1) > A(n, d, w) with w ≤ n/2. A scheme based on double counting is developed for validating the classification results. [ABSTRACT FROM AUTHOR]

Let A(n,d) denote the maximum size of a binary code with length n and minimum distance d. It has been known for decades that A(16,7) = A(17,8) = 36 or 37, that is, that the size of optimal 3-error-correcting binary codes of length 16 is either 36 or 37. By a recursive classification via subcodes and a clique search in the final stage, it is shown that the size of optimal such codes is 36. [ABSTRACT FROM AUTHOR]