Evaluation codes from order domain theory
Finite Fields and Their Applications
A new method for constructing small-bias spaces from hermitian codes
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
Weighted Reed---Muller codes revisited
Designs, Codes and Cryptography
| Hi-index | 754.84 |
For any nonzero elementcof a general finite fieldGF(q), it is shown that the polynomials(x - c)^i, i = 0,1,2,cdots, have the "weight-retaining" property that any linear combination of these polynomials with coefficients inGF(q)has Hamming weight at least as great as that of the minimum degree polynomial included. This fundamental property is then used as the key to a variety of code constructions including 1) a simplified derivation of the binary Reed-Muller codes and, for any primepgreater than 2, a new extensive class ofp-ary "Reed-Muller codes," 2) a new class of "repeated-root" cyclic codes that are subcodes of the binary Reed-Muller codes and can be very simply instrumented, 3) a new class of constacyclic codes that are subcodes of thep-ary "Reed-Muller codes," 4) two new classes of binary convolutional codes with large "free distance" derived from known binary cyclic codes, 5) two new classes of long constraint length binary convolutional codes derived from2^r-ary Reed-Solomon codes, and 6) a new class ofq-ary "repeated-root" constacyclic codes with an algebraic decoding algorithm.