57 research outputs found
On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes
We derive the Wu list-decoding algorithm for Generalised Reed-Solomon (GRS)
codes by using Gr\"obner bases over modules and the Euclidean algorithm (EA) as
the initial algorithm instead of the Berlekamp-Massey algorithm (BMA). We
present a novel method for constructing the interpolation polynomial fast. We
give a new application of the Wu list decoder by decoding irreducible binary
Goppa codes up to the binary Johnson radius. Finally, we point out a connection
between the governing equations of the Wu algorithm and the Guruswami-Sudan
algorithm (GSA), immediately leading to equality in the decoding range and a
duality in the choice of parameters needed for decoding, both in the case of
GRS codes and in the case of Goppa codes.Comment: To appear in IEEE Transactions of Information Theor
Subquadratic time encodable codes beating the Gilbert-Varshamov bound
We construct explicit algebraic geometry codes built from the
Garcia-Stichtenoth function field tower beating the Gilbert-Varshamov bound for
alphabet sizes at least 192. Messages are identied with functions in certain
Riemann-Roch spaces associated with divisors supported on multiple places.
Encoding amounts to evaluating these functions at degree one places. By
exploiting algebraic structures particular to the Garcia-Stichtenoth tower, we
devise an intricate deterministic \omega/2 < 1.19 runtime exponent encoding and
1+\omega/2 < 2.19 expected runtime exponent randomized (unique and list)
decoding algorithms. Here \omega < 2.373 is the matrix multiplication exponent.
If \omega = 2, as widely believed, the encoding and decoding runtimes are
respectively nearly linear and nearly quadratic. Prior to this work, encoding
(resp. decoding) time of code families beating the Gilbert-Varshamov bound were
quadratic (resp. cubic) or worse
Understanding binary-Goppa decoding
This paper reviews, from bottom to top, a polynomial-time algorithm to correct t errors in classical binary Goppa codes defined by squarefree degree-t polynomials. The proof is factored through a proof of a simple Reed--Solomon decoder, and the algorithm is simpler than Patterson\u27s algorithm. All algorithm layers are expressed as Sage scripts. The paper also covers the use of decoding inside the Classic McEliece cryptosystem, including reliable recognition of valid inputs
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