5 research outputs found
On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes
Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point
codes leads to two methods for improving code rate. One method, due to Feng and
Rao, removes parity checks that may be recovered by their majority voting
algorithm. The second method is to design the code to correct only those error
vectors of a given weight that are also geometrically generic. In this work,
formulae are given for the redundancies of Hermitian codes optimized with
respect to these criteria as well as the formula for the order bound on the
minimum distance. The results proceed from an analysis of numerical semigroups
generated by two consecutive integers. The formula for the redundancy of
optimal Hermitian codes correcting a given number of errors answers an open
question stated by Pellikaan and Torres in 1999.Comment: Added reference
Improved Two-Point Codes on Hermitian Curves
One-point codes on the Hermitian curve produce long codes with excellent
parameters. Feng and Rao introduced a modified construction that improves the
parameters while still using one-point divisors. A separate improvement of the
parameters was introduced by Matthews considering the classical construction
but with two-point divisors. Those two approaches are combined to describe an
elementary construction of two-point improved codes. Upon analysis of their
minimum distance and redundancy, it is observed that they improve on the
previous constructions for a large range of designed distances
Unique Decoding of Plane AG Codes via Interpolation
We present a unique decoding algorithm of algebraic geometry codes on plane
curves, Hermitian codes in particular, from an interpolation point of view. The
algorithm successfully corrects errors of weight up to half of the order bound
on the minimum distance of the AG code. The decoding algorithm is the first to
combine some features of the interpolation based list decoding with the
performance of the syndrome decoding with majority voting scheme. The regular
structure of the algorithm allows a straightforward parallel implementation.Comment: Submitted for publication in the Transactions on Information Theor
Coset bounds for algebraic geometric codes
For a given curve X and divisor class C, we give lower bounds on the degree
of a divisor A such that A and A-C belong to specified semigroups of divisors.
For suitable choices of the semigroups we obtain (1) lower bounds for the size
of a party A that can recover the secret in an algebraic geometric linear
secret sharing scheme with adversary threshold C, and (2) lower bounds for the
support A of a codeword in a geometric Goppa code with designed minimum support
C. Our bounds include and improve both the order bound and the floor bound. The
bounds are illustrated for two-point codes on general Hermitian and Suzuki
curves.Comment: 36 page
On semigroups generated by two consecutive integers and improved Hermitian codes
Analysis of the Berlekamp-Massey-Sakata algorithm for decoding onepoint codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999