Construction of New MDS Codes from Gabidulin Codes
Thierry Pierre Berger, A.V. Ourivski · 2004
In this paper, we present a method to construct new MDS codes by extending Gabidulin codes. This method uses optimal codes on the underlying base field. 1 Rank distance and Gabidulin codes 1.1 Rank distance The rank distance was introduced by E. Gabidulin in 1985. For more details on this metric the reader is referred to [1]. We just recall the results needed in this paper. Let K = GF (qm) be an extension of degree m of the finite field GF (q). Note that q = pr is not necessary a prime, however, the field GF (q) is considered as the “base field ” in this paper. Let E = Kn be the vector space of dimension n over K. Definition 1 For a ∈ E, a = (a1,..., an), the rank weight rk(a) of a is the dimension of the GF (q)-vector space generated by {a1,..., an}. Equivalently, the rank weight of a is the rank of the m × n-matrix over GF (q) formed by extending every coordinate ai on a basis of K/GF (q). The value of rank weight is obviously independent of the chosen basis. For example, let K = GF (23) = GF (2)(α) with α3 = α+ 1, and n = 5. Set x = (α6,α, 0,α5,α) = (α2 + 1,α, 0,α2 + α+ 1,α). the rank of x is rk(x) = 2. It is also the rank of the matrix