Optimal Codes with Positive Griesmer Defects, Related Optimal and Almost Optimal LRC Codes
For coding theorists, this provides new constructions of optimal codes beyond known families, addressing a challenging problem in code optimality.
The authors construct infinite families of optimal linear codes with positive Griesmer defects, which are not equivalent to known Solomon-Stiffler or Belov codes, and determine their weight distributions. Some of these codes are optimal or almost optimal locally recoverable codes (LRCs) meeting or approaching the Cadambe-Mazumdar bound with locality two.
Solomon and Stiffler constructed infinitely many families of linear codes meeting the Griesmer bound in 1965. It is well-known in 1990's that certain Griesmer codes (codes with the zero Griesmer defect) are equivalent to Solomon-Stiffler codes or Belov codes. Griesmer codes constructed in some recent papers published in IEEE Trans. Inf. Theory are actually Solomon-Stiffler codes or affine Solomon-Stiffler codes proposed in our previous paper. Therefore it is more challenging to construct optimal codes with positive Griesmer defects. In this paper, we construct several infinite families of optimal codes with positive Griesmer defects. Then these codes are certainly not equivalent to Solomon-Stiffler codes or Belov codes. Weight distributions and subcode support weight distributions of these optimal codes are determined. On the other hand, some of constructed optimal linear codes are optimal locally recoverable codes (LRCs) meeting the Cadambe-Mazumdar (CM) bound. Some of our constructed optimal codes are very close to the CM bound. Localities of these optimal or almost optimal LRC codes are two.