An algebraic approach for decoding spread codes

Abstract

In this paper we study spread codes: a family of constant-dimension codes for random linear network coding. In other words, the codewords are full-rank matrices of size k × n with entries in a finite field Fq. Spread codes are a family of optimal codes with maximal minimum distance. We give a minimum-distance decoding algorithm which requires O((n − k)k 3 ) operations over an extension field F q k . Our algorithm is more efficient than the previous ones in the literature, when the dimension k of the codewords is small with respect to n. The decoding algorithm takes advantage of the algebraic structure of the code, and it uses original results on minors of a matrix and on the factorization of polynomials over finite fields.

Subspace codes from Ferrers diagrams

“…Remark 55. Spread and partial spread codes ( [13], [6], [7]) can be obtained through the multilevel construction for a special choice of the pivot vectors. The Ferrers diagrams associated to those pivot vectors are studied in Theorem 15.…”

Message encoding for spread and orbit codes

“…They exist if and only if k|n, have minimum distance 2k and cardinality (q n − 1)/(q k − 1). For more information on different constructions and decoding algorithms of spread codes see [8], [12], [13], [20]. We will use the following construction, which gives rise to a Desarguesian spread code in G q (k, n) ([20]): 1) Let m := n/k and consider G q k (1, m), which has q k(m−1) +q k(m−2) +q k(m−3) +· · ·+1 = (q n −1)/(q k −1) elements.…”

Isometry and automorphisms of constant dimension codes

“…Different constructions for these codes are known and have been studied from a coding perspective, e.g. in [8,13,14].…”