Optimal, Almost Optimal Few-Weight Linear Codes and Related Quantum Codes

In eight published papers in IEEE Transactions on Information Theory, infinite families of optimal few-weight binary and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>-ary linear codes were constructed and their weight distributions were determined. These codes are linear codes meeting the Griesmer bound. We indicate that many Griesmer codes constructed in these papers are not new. They are actually Solomon-Stiffler codes invented in 1965. Therefore weight distributions of some special binary or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>-ary Solomon-Stiffler codes were determined in the papers mentioned above. From a similar geometric approach as Solomon-Stiffler codes, we construct ten infinite families of binary, ternary and quaternary few-weight, optimal, almost optimal and near-optimal linear codes close to the Griesmer bound and their weight distributions are determined. These linear codes have positive Griesmer defects up to five, and thus not Solomon-Stiffler codes and Griesmer codes from minihypers. Moreover, many optimal, best known and almost optimal quantum codes of small lengths, comparing with Grassl's table on quantum codes, are constructed from the same geometric approach as binary Solomon-Stiffler codes.