144,999 research outputs found
On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes
In this paper, we present an improved union bound on the Linear Programming
(LP) decoding performance of the binary linear codes transmitted over an
additive white Gaussian noise channels. The bounding technique is based on the
second-order of Bonferroni-type inequality in probability theory, and it is
minimized by Prim's minimum spanning tree algorithm. The bound calculation
needs the fundamental cone generators of a given parity-check matrix rather
than only their weight spectrum, but involves relatively low computational
complexity. It is targeted to high-density parity-check codes, where the number
of their generators is extremely large and these generators are spread densely
in the Euclidean space. We explore the generator density and make a comparison
between different parity-check matrix representations. That density effects on
the improvement of the proposed bound over the conventional LP union bound. The
paper also presents a complete pseudo-weight distribution of the fundamental
cone generators for the BCH[31,21,5] code
Minimal binary linear codes - a general framework based on bent concatenation
Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function in the so-called direct sum of Boolean functions , where is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length and dimension (assuming that h: \F_2^n \rightarrow \F_2), whose weight distribution is exactly specified for certain choices of . To increase the dimension of these codes with respect to their length, we introduce the concept of \textit{non-covering permutations} (referring to the property of minimality) used to construct a bent function in variables, which allows us to employ a suitable subspace of derivatives of and generate minimal codes of dimension instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal linear codes that cross the so-called Ashikhmin-Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension , through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin-Barg bound. More precisely, for a suitable choice of derivatives of , where is a bent function and satisfies certain minimality requirements, for any fixed , one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation when specifying the bent function in the Maiorana-McFarland class. The weight distribution is given explicitly for any (suitable) when is an almost bent permutation
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