38 research outputs found
A Note on “A Systematic (12,8) Code for Correcting Single Errors and Detecting Adjacent Errors”
J.W. Schwartz and J.K. Wolf (ibid., vol. 39, no. 11, pp. 1403-1404, Nov. 1990) gave a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects eight of the nine double adjacent errors within any of the three 4-bit nibbles. We present a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects any pair of errors within a nibble
Interpolation and Approximation of Polynomials in Finite Fields over a Short Interval from Noisy Values
Motivated by a recently introduced HIMMO key distribution scheme, we consider
a modification of the noisy polynomial interpolation problem of recovering an
unknown polynomial from approximate values of the residues of
modulo a prime at polynomially many points taken from a short
interval
On parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size
Recently there has been interest in the construction of small parity check
sets for iterative decoding of the Hamming code with the property that each
uncorrectable (or stopping) set of size three is the support of a codeword and
hence uncorrectable anyway. Here we reformulate and generalise the problem, and
improve on this construction. First we show that a parity check collection that
corrects all correctable erasure patterns of size m for the r-th order Hamming
code (i.e, the Hamming code with codimension r) provides for all codes of
codimension a corresponding ``generic'' parity check collection with this
property. This leads naturally to a necessary and sufficient condition on such
generic parity check collections. We use this condition to construct a generic
parity check collection for codes of codimension r correcting all correctable
erasure patterns of size at most m, for all r and m <= r, thus generalising the
known construction for m=3. Then we discussoptimality of our construction and
show that it can be improved for m>=3 and r large enough. Finally we discuss
some directions for further research.Comment: 13 pages, no figures. Submitted to IEEE Transactions on Information
Theory, July 28, 200
Entropy of a bit-shift channel
We consider a simple transformation (coding) of an iid source called a
bit-shift channel. This simple transformation occurs naturally in magnetic or
optical data storage. The resulting process is not Markov of any order. We
discuss methods of computing the entropy of the transformed process, and study
some of its properties.Comment: Published at http://dx.doi.org/10.1214/074921706000000293 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org