4,116 research outputs found
Spectrum of Sizes for Perfect Deletion-Correcting Codes
One peculiarity with deletion-correcting codes is that perfect
-deletion-correcting codes of the same length over the same alphabet can
have different numbers of codewords, because the balls of radius with
respect to the Levenshte\u{\i}n distance may be of different sizes. There is
interest, therefore, in determining all possible sizes of a perfect
-deletion-correcting code, given the length and the alphabet size~.
In this paper, we determine completely the spectrum of possible sizes for
perfect -ary 1-deletion-correcting codes of length three for all , and
perfect -ary 2-deletion-correcting codes of length four for almost all ,
leaving only a small finite number of cases in doubt.Comment: 23 page
Super-simple directed designs and their smallest defining sets with its application in LDPC codes
In this paper, we show that for all (mod 5) and ,
there exists a super-simple directed design, also for these
parameters there exists a super-simple directed design such that its
smallest defining sets contain at least half of its blocks. Also, we show that
these designs are useful in constructing parity-check matrices of LDPC codes.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0009
A lower bound on HMOLS with equal sized holes
It is known that , the maximum number of mutually orthogonal latin
squares of order , satisfies the lower bound for large
. For , relatively little is known about the quantity ,
which denotes the maximum number of `HMOLS' or mutually orthogonal latin
squares having a common equipartition into holes of a fixed size . We
generalize a difference matrix method that had been used previously for
explicit constructions of HMOLS. An estimate of R.M. Wilson on higher
cyclotomic numbers guarantees our construction succeeds in suitably large
finite fields. Feeding this into a generalized product construction, we are
able to establish the lower bound for any
and all
Intriguing sets of strongly regular graphs and their related structures
In this paper we outline a technique for constructing directed strongly regular graphs by using strongly regular graphs having a "nice" family of intriguing sets. Further, we investigate such a construction method for rank three strongly regular graphs having at most vertices. Finally, several examples of intriguing sets of polar spaces are provided
Existence of r-fold perfect (v,K,1)-Mendelsohn designs with K⊆{4,5,6,7}
AbstractLet v be a positive integer and let K be a set of positive integers. A (v,K,1)-Mendelsohn design, which we denote briefly by (v,K,1)-MD, is a pair (X,B) where X is a v-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t=1,2,…,r, every ordered pair of points of X are t-apart in exactly one block of B, then the (v,K,1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect (v,K,1)-MD. If K={k} and r=k−1, then an r-fold perfect (v,{k},1)-MD is essentially the more familiar (v,k,1)-perfect Mendelsohn design, which is briefly denoted by (v,k,1)-PMD. In this paper, we investigate the existence of r-fold perfect (v,K,1)-Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements
- …