Linear codes are considered. A code is characterized by the length n, the dimension k, and the minimum distance d. An (n,k,d) code over the finite field GF(q) is said to be maximum distance separable (MDS) if d=n-k+1. A. Krishna and D.V. Sarwate (1990) investigated the existence of pseudo-cyclic MDS codes over GF(q) of length n, where n divides q-1 or q+1. It is shown that the pseudo-cyclic MDS codes constructed by Krishna and Sarwate are generalised Reed-Solomon codes. Pseudo-cyclic codes are studied over GF(q), where n and q=p/sup h/ are not relatively prime. It is proven that pseudo-cyclic (n,k) MDS codes modulo (x/sup n/-a) over GF(q) exist, if and only if n= p. Furthermore, any pseudo-cyclic (p,k) code modulo (x/sup p/-a) over GF(q) turns out to be MDS and generalized Reed-Solomon. It is explicitly proven that some classes of pseudo-cyclic (n,k) MDS codes over GF(q) are generalized Reed-Solomon codes. Furthermore, pseudo-cyclic (q+1,4) MDS codes over GF(q), q even, are completely classified.
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机译：考虑线性代码。代码的特征在于长度n，尺寸k和最小距离d。如果d = n-k + 1，则在有限域GF（q）上的（n，k，d）码被称为最大距离可分离（MDS）。克里希纳和D.V. Sarwate（1990）研究了长度为n的GF（q）上伪循环MDS代码的存在，其中n划分q-1或q + 1。结果表明，克里希纳和萨尔瓦特构造的伪循环MDS码是广义的里德-所罗门码。在GF（q）上研究伪循环码，其中n和q = p / sup h /不是相对质数。已经证明，当且仅当n = p时，存在以GF（q）为模（x / sup n / -a）的伪循环（n，k）MDS代码。此外，在GF（q）上模（x / sup p / -a）的任何伪循环（p，k）码都证明是MDS和广义Reed-Solomon。明确证明，GF（q）上的某些类伪循环（n，k）MDS代码是广义的Reed-Solomon代码。此外，GF（q）甚至q上的伪循环（q + 1,4）MDS代码也被完全分类。
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