Relative generalized Hamming weights of one-point algebraic geometric codes

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights

Optimal Threshold-Based Multi-Trial Error/Erasure Decoding with the Guruswami-Sudan Algorithm

Traditionally, multi-trial error/erasure decoding of Reed-Solomon (RS) codes is based on Bounded Minimum Distance (BMD) decoders with an erasure option. Such decoders have error/erasure tradeoff factor L=2, which means that an error is twice as expensive as an erasure in terms of the code's minimum distance. The Guruswami-Sudan (GS) list decoder can be considered as state of the art in algebraic decoding of RS codes. Besides an erasure option, it allows to adjust L to values in the range 1<L<=2. Based on previous work, we provide formulae which allow to optimally (in terms of residual codeword error probability) exploit the erasure option of decoders with arbitrary L, if the decoder can be used z>=1 times. We show that BMD decoders with z_BMD decoding trials can result in lower residual codeword error probability than GS decoders with z_GS trials, if z_BMD is only slightly larger than z_GS. This is of practical interest since BMD decoders generally have lower computational complexity than GS decoders.Comment: Accepted for the 2011 IEEE International Symposium on Information Theory, St. Petersburg, Russia, July 31 - August 05, 2011. 5 pages, 2 figure

On the zeros and poles of a transfer function

AbstractThe poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module. These algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function. In addition, these spaces carry the structure of a module over a ring of polynomials, which gives them a dynamical or state space structure. The analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions. Following ideas of Wedderburn and Forney, we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective. A new exact sequence relates the global spaces of zeros, the global spaces of poles, and the new generic zero spaces. This sequence gives a structural result which can be summarized as follows: “The number of zeros of any transfer function is equal to the number of poles (when everything is counted appropriately).” The same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general (possibly improper) transfer functions to controlled invariant and controllability subspaces (including such spaces at infinity)

Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems