2,127 research outputs found
Weight distribution of some reducible cyclic codes
AbstractLet q=pm where p is an odd prime, m⩾3, k⩾1 and gcd(k,m)=1. Let Tr be the trace mapping from Fq to Fp and ζp=e2πip. In this paper we determine the value distribution of following two kinds of exponential sums∑x∈Fqχ(αxpk+1+βx2)(α,β∈Fq) and∑x∈Fqχ(αxpk+1+βx2+γx)(α,β,γ∈Fq), where χ(x)=ζpTr(x) is the canonical additive character of Fq. As an application, we determine the weight distribution of the cyclic codes C1 and C2 over Fp with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where h1(x), h2(x) and h3(x) are the minimal polynomials of π−1, π−2 and π−(pk+1) over Fp, respectively, for a primitive element π of Fq
Two classes of reducible cyclic codes with large minimum symbol-pair distances
The high-density data storage technology aims to design high-capacity storage
at a relatively low cost. In order to achieve this goal, symbol-pair codes were
proposed by Cassuto and Blaum \cite{CB10,CB11} to handle channels that output
pairs of overlapping symbols. Such a channel is called symbol-pair read
channel, which introduce new concept called symbol-pair weight and minimum
symbol-pair distance. In this paper, we consider the parameters of two classes
of reducible cyclic codes under the symbol-pair metric. Based on the theory of
cyclotomic numbers and Gaussian period over finite fields, we show the possible
symbol-pair weights of these codes. Their minimum symbol-pair distances are
twice the minimum Hamming distances under some conditions. Moreover, we obtain
some three symbol-pair weight codes and determine their symbol-pair weight
distribution. A class of MDS symbol-pair codes is also established. Among other
results, we determine the values of some generalized cyclotomic numbers
A Class of Three-Weight Cyclic Codes
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, a class of
three-weight cyclic codes over \gf(p) whose duals have two zeros is
presented, where is an odd prime. The weight distribution of this class of
cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a
subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page
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