3,490 research outputs found
Codes and finite geometries
We explore the connections between finite geometry and algebraic coding theory, giving a rather full account of the Reed-Muller and generalized Reed-Muller codes. Some of the results and many of the proofs are new but this is largely an expository effort that relies heavily on the work of Delsarte et al. and of Charpin. The necessary geometric background is sketched before we begin the discussion of the Reed-Muller codes and their p-ary analogues. We prove all the classical results concerning these codes and include a discussion of the group-algebra approach and prove Berman's theorem characterizing the codes as powers of the radical. Included also is a discussion of the characterization of affine-invariant cyclic codes given by Kasami, Lin and Peterson and its generalization by Delsarte. our theme throughout this work is the relationship between these codes and the codes coming from both affine and projective geometries. The final section develops the theory in the more difficult case in which the field is not of prime order, here must look at subfield subcodes - which complicates the connection with the geometric codes, which are codesover the prime subfield of the field of the geometry
Optimal Testing of Generalized Reed-Muller Codes in Fewer Queries
A local tester for an error correcting code is a
tester that makes oracle queries to a given word and
decides to accept or reject the word . An optimal local tester is a local
tester that has the additional properties of completeness and optimal
soundness. By completeness, we mean that the tester must accept with
probability if . By optimal soundness, we mean that if the tester
accepts with probability at least (where is small),
then it must be the case that is -close to some codeword
in Hamming distance.
We show that Generalized Reed-Muller codes admit optimal testers with queries. Here, for a prime power , the Generalized Reed-Muller code, RM[n,q,d], consists of the
evaluations of all -variate degree polynomials over .
Previously, no tester achieving this query complexity was known, and the best
known testers due to Haramaty, Shpilka and Sudan(which is optimal) and due to
Ron-Zewi and Sudan(which was not known to be optimal) both required
queries. Our tester achieves query
complexity which is polynomially better than by a power of , which is
nearly the best query complexity possible for generalized Reed-Muller codes.
The tester we analyze is due to Ron-Zewi and Sudan, and we show that their
basic tester is in fact optimal. Our methods are more general and also allow us
to prove that a wide class of testers, which follow the form of the Ron-Zewi
and Sudan tester, are optimal. This result applies to testers for all
affine-invariant codes (which are not necessarily generalized Reed-Muller
codes).Comment: 42 pages, 8 page appendi
The -Number of Binomial Edge Ideals
The invariant -number was introduced very recently in the study
of Reed-Muller-type codes. Jaramillo and Villarreal (J Combin. Theory Ser. A
177:105310, 2021) initiated the study of the -number of edge
ideals. Inspired by their work, we take the initiation to study the
-number of binomial edge ideals in this paper. We discuss some
properties and bounds of the -number of binomial edge ideals. We
explicitly find the -number of binomial edge ideals locally at the
associated prime corresponding to the cutset . We show that the
-number of Knutson binomial edge ideals is less than or equal to
the -number of their initial ideals. Also, we classify all binomial
edge ideals whose -number is . Moreover, we try to relate the
-number with the Castelnuvo-Mumford regularity of binomial edge
ideals and give a conjecture in this direction
Symmetries of weight enumerators and applications to Reed-Muller codes
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a
crucial role for research in coding theory during the last four decades. Plenty
of generalizations have been proved but, to our knowledge, they are all based
on the symmetries given by MacWilliams' identities. This paper is intended to
be a first step towards a more general investigation of symmetries of weight
enumerators. We list the possible groups of symmetries, dealing both with the
finite and infinite case, we develop a new algorithm to compute the group of
symmetries of a given weight enumerator and apply these methods to the family
of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's
theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To
appear in Advances in Mathematics of Communication
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