| dc.description.abstract |
GF(q) denote the finite field with q elements. An [n,k,d] linear code C over
GF(q) is a k-dimensional subspace of GF(q) ^n with minimum (Hamming) distance d.
The vectors in C are codewords of C. Specially, codes over GF(2) are called binary
linear codes.
Let q=2^m for some positive integer m, and Tr denote the trace map from
GF(q) onto GF(2). For D={d_1,d_2,...,d_n} subset GF(q)^* , we define a linear code of length
n over GF(2) by
C_D={Tr(xd_1),Tr(xd_2),...,Tr(xd_1): x in GF(q) },
and call D the defining set of this code C_D. This construction approach of the linear
codes was employed by Cunsheng Dings in [1] and [2] for obtaining linear codes
with a few weights. Different orderings of the elements of Dresult in different codes
C_D, but the codes are permutation equivalent.
A Hamming code is a linear code for error detection that can detect up to two
simultaneous bit errors and is capable of correcting single-bit errors. The duals of the
Hamming codes are simplex codes. A code is called constant-weight code if all
nonzero codewords of this code have the same weight. The simplex codes are
constant-weight codes. Reed-Muller codes are amongst the oldest and most wellknown
of codes. All codewords, except (0,0,...,0) and (1,1,...,1) codewords, of the first
order Reed-Muller code have the same weight.
The objective of this study is to produce the known two binary linear codes by
selecting the defining set D subset GF(q)^* . The first code is the binary simplex code. The
second code is the first order Reed-Muller code. |
tr_TR |