Subspace Subcodes of Reed-Solomon Codes

— In this paper we introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed–Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed–Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed -dimensional vector subspace S of GF (2 m ): SSRS codes are constructed using properties of the Galois field GF (2 m ): They are not linear over the field GF (2  ), which does not come into play, but rather are Abelian group codes over S: However, they are linear over GF (2), and the symbolwise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that 1) SSRS codes can have a higher dimension tha...

• Type: Journal
• Year: 1998
• Where: TIT
• Authors: Masayuki Hattori, Robert J. McEliece, Gustave Solomon
• Keywords: -dimensional Vector Subspace, Cyclic Error-correcting Codes, Ssrs Codes, TIT 1998