242,227 research outputs found
The weight enumerator polynomials of the lifted codes of the projective Solomon-Stiffler codes
Determining the weight distribution of a code is an old and fundamental topic
in coding theory that has been thoroughly studied. In 1977, Helleseth,
Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the
lifted code over of a -ary linear code with
significant combinatorial properties, which can determine the support weight
distribution of this linear code. The Solomon-Stiffler codes are a family of
famous Griesmer codes, which were proposed by Solomon and Stiffler in 1965. In
this paper, we determine the weight enumerator polynomials of the lifted codes
of the projective Solomon-Stiffler codes using some combinatorial properties of
subspaces. As a result, we determine the support weight distributions of the
projective Solomon-Stiffler codes. In particular, we determine the weight
hierarchies of the projective Solomon-Stiffler codes.Comment: This manuscript was first submitted on September 9, 202
The problem with the SURF scheme
There is a serious problem with one of the assumptions made in the security
proof of the SURF scheme. This problem turns out to be easy in the regime of
parameters needed for the SURF scheme to work.
We give afterwards the old version of the paper for the reader's convenience.Comment: Warning : we found a serious problem in the security proof of the
SURF scheme. We explain this problem here and give the old version of the
paper afterward
Near MDS poset codes and distributions
We study -ary codes with distance defined by a partial order of the
coordinates of the codewords. Maximum Distance Separable (MDS) codes in the
poset metric have been studied in a number of earlier works. We consider codes
that are close to MDS codes by the value of their minimum distance. For such
codes, we determine their weight distribution, and in the particular case of
the "ordered metric" characterize distributions of points in the unit cube
defined by the codes. We also give some constructions of codes in the ordered
Hamming space.Comment: 13 pages, 1 figur
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