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In this paper we present several values for the next-to-minimal weights of projective Reed–Muller codes. We work over F q with q ≥ 3 since in [3] we have determined the complete values for the next-to-minimal weights of binary projective Reed–Muller codes. As in [3] here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } . [ABSTRACT FROM AUTHOR]

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. [ABSTRACT FROM AUTHOR]

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. Let m = 2 ℓ + 1 for an integer ℓ ≥ 1 and π be a generator of GF ( 3 m ) ⁎ . In this paper, a class of cyclic codes C ( u , v ) over GF ( 3 ) with two nonzeros π u and π v is studied, where u = ( 3 m + 1 ) / 2 , and v = 2 ⋅ 3 ℓ + 1 is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF ( 3 m ) , the cyclic code C ( u , v ) is shown to have minimal distance four, which is the best minimal distance for any linear code over GF ( 3 ) with length 3 m − 1 and dimension 3 m − 1 − 2 m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied. [ABSTRACT FROM AUTHOR]