The paper has a threefold purpose. The first purpose is to present an explicit description of expanded cyclic codes defined in GF(qm). The proposed explicit construction of expanded generator matrix and expanded parity check matrix maintains the symbol-wise algebraic structure and thus keeps many important original characteristics. The second purpose of this paper is to identify a class of cons...

The McEliece cryptosystem is one of the oldest public-key cryptosystem ever designated. It is also the first public-key cryptosystem based on linear error-correcting codes. The main advantage of the McEliece cryptosystem is to have a very fast encryption and decryption functions but suffers from a major drawback. It requires a very large public key which makes it very difficult to use in many p...

In this paper, cyclic codes are established over some finite quaternion integer rings with respect to the quaternion Mannheim distance, and decoding algorithm for these codes is given. 2000 AMS Classification: 94B05, 94B15, 94B35, 94B60

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of Tr(x) over F2m in the case wherem = 2t, d = 3+2 t+1 and gcd(d, 2−1) = 1.

or respectively 2 m−e 2e )d + 1, 2 m−3e 2e )d + 1, · · · , 2 m+3e 2e −k)d + 1. Then s1, · · · , sk−1 are maximum-length binary sequences of length 2 m − 1. Let C be the F2-vector space generated by all circular shifts of s, s1, · · · , sk−1. Then C has an F2m-vector space structure, and is of dimension k over F2m . When k = 2, C is the Gold code. So we regard C as a Gold type code of relative d...

In this paper, we investigate cyclic codes over the ring Fp[u, v, w]〈u , v, w, uv− vu, vw − wv, uw − wu〉, where p is a prime number. Which is a part of family of Frobenius rings. We find a unique set of generators for these codes and characterize the free cyclic codes. We also study the rank and the Hamming distance of these codes. We also constructs some good p−ary codes as the Gray images of ...

Additive cyclic codes over Galois rings were investigated in [3]. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the study in [3], whereas the other one has some unusual propertie...

The ring in the title is the first non commutative ring to have been used as alphabet for block codes. The original motivation was the construction of some quaternionic modular lattices from codes. The new application is the construction of space time codes obtained by concatenation from the Golden code. In this article, we derive structure theorems for cyclic codes over that ring, and use them...

Cyclic codes have been an important topic of both mathematics and engineering for decades. They have beenwidely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic codes and their weight distributions. Informati...