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In coding theory, Singleton bound is an upper bound on the size of an arbitrary linear code with length n, size M, and minimum distance d Let C be an [n, k, d] code. The Singleton Bound says that d ≤ n – k + 1. If this equality holds, or d = n – k + 1, then the code is called MDS code or Maximum Distance separable code.

 

One of the most commonly used MDS code is the Generalized Reed Solomon (GRS) code. Let n be an arbitrary integer such that 1 ≤ n ≤ q. Let a=(a_1, a_2, …, a_n) be an n-tuple of mutually distinct elements of ð”½_q ∪{∞}. Let b be an n-tuple of nonzero elements of ð”½_q Let k be an arbitrary integer such that 0 ≤ k ≤ n. C is a generalized Reed-Solomon code, denoted by GRS_k(a,b), is generated by EQUATION (1) [PLEASE SEE THE ABSTRACT FILE]

 

One of the decoding methods for linear codes is error-correcting pairs. Let C be an ð”½_q -linear code of length n. Let A, B be subspaces of ð”½_(q^m)^n. Then (A, B) is called a t-error-correcting pair (t-ECP) for C if the following conditions are satisfied:

(1)        (A * B)⊥C,

(2)        k(A) > t,

(3)        d(B^⊥) > t,

(4)       d(A)+d(C) > n.

In 2015, Márquez-Corbella and Pellikan characterize MDS code that has an error-correcting pair. The result shows that an MDS code of minimum distance 2t + 1 has a t-ECP if and only if it is a generalized Reed-Solomon code.

 

A set S ⊆ ð”½_q of size m is called an (m, t, \delta)-set in ð”½_q if there exists an element \delta \in ð”½_q such that no t elements of S sum to \delta. Let n and k be two integers such that k ≥ 3 and k+3 ≤ n ≤ q+2. Let C be an [n, k] code over ð”½_q that is generated by EQUATION (2) [PLEASE SEE THE ABSTRACT FILE]

 

where the \alpha_i are distinct elements of ð”½_q and \delta \in ð”½_q. Then G does not generate a GRS code, and G generates an MDS code if and only if the \alpha_i form an (n – 2, k – 1, \delta)-set in ð”½_q. In this paper, we will show that if B is an [n, 1, n] code that is generated by EQUATION (3) [PLEASE SEE THE ABSTRACT FILE]

 

than (C^⊥, B) is a 1-ECP  for C.

 

MDS Code; Generalized Reed-Solomon (GRS) Code; Error-Correcting Pair