26 research outputs found
Construction of sequences with high Nonlinear Complexity from the Hermitian Function Field
We provide a sequence with high nonlinear complexity from the Hermitian
function field over . This sequence was
obtained using a rational function with pole divisor in certain
collinear rational places on , where . In
particular we improve the lower bounds on the th-order nonlinear complexity
obtained by H. Niederreiter and C. Xing; and O. Geil, F. \"Ozbudak and D.
Ruano
Hermitian codes from higher degree places
Matthews and Michel investigated the minimum distances in certain
algebraic-geometry codes arising from a higher degree place . In terms of
the Weierstrass gap sequence at , they proved a bound that gives an
improvement on the designed minimum distance. In this paper, we consider those
of such codes which are constructed from the Hermitian function field. We
determine the Weierstrass gap sequence where is a degree 3 place,
and compute the Matthews and Michel bound with the corresponding improvement.
We show more improvements using a different approach based on geometry. We also
compare our results with the true values of the minimum distances of Hermitian
1-point codes, as well as with estimates due Xing and Chen
Applications of Algebraic Geometric Codes to Polar Coding
In recent groundbreaking work, Arikan developed polar codes as an explicit construction of symmetric capacity achieving codes for binary discrete memoryless channels with low encoding and decoding complexities. In this construction, a specific kernel matrix G is considered and is used to encode a block of channels. As the number of channels grows, each channel becomes either a noiseless channel or a pure-noise channel, and the rate of this polarization is related to the kernel matrix used. Since Arikan\u27s original construction, polar codes have been generalized to q-ary discrete memoryless channels, where q is a power of a prime, and other matrices have been considered as kernels. In our work, we expand on the ideas of Mori and Tanaka and Korada, Sasoglu, and Urbanke by employing algebraic geometric codes to produce kernels of polar codes, specifically codes from maximal and optimal function fields