Family Of Binary Codes Research Articles

Binary Linear Codes With Optimal Scaling: Polar Codes With Large Kernels

We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within ε > 0 of capacity, the code length n often scales as O(1/ε <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">μ</sup> ), where the constant μ is called the scaling exponent. It is known that the optimal scaling exponent is μ = 2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2×2 kernel) on the BEC is μ = 3.63. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> × <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> binary kernels, such that polar codes constructed from these kernels achieve scaling exponent μ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> ) that tends to the optimal value of 2 as <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> grows. We furthermore characterize precisely how large <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> needs to be as a function of the gap between μ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> ) and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity O(n) and encoding/decoding complexity O(nlogn).

On involutions in extremal self-dual codes and the dual distance of semi self-dual codes

A classical result of Conway and Pless is that a natural projection of the fixed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual [13]. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes.In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes.As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and σ∈Aut(C) be a fixed point free automorphism of order 2. If m is odd or if m=2k with (5k−1k−1) odd then C is a free F2〈σ〉-module. This result has quite strong consequences on the structure of the automorphism group of such codes.

Codes Derived from Binary Goppa Codes

We present a new family of binary codes derived from the family of classical Goppa codes. We generalize properties of Goppa codes to this family, including bounds on the dimension and minimum distance and construction of a polynomial-time algorithm of decoding up to the designed distance. Asymptotically, these codes have the same parameters as Goppa codes.

Deterministic tests for detecting singleV-coupling faults in RAMs

We consider the problem of detecting singleV-coupling faults (as defined by Nair, Thatte, and Abraham) inn×1 random-access memories (RAMs). First we derive a lower bound of 2 V−2 nlog2 n+(2 V +3)n on the length of any test that detects all singleV-coupling faults, for 2≤V≤47 andn=2 e whereeɛ{8,...,34}. In the derivation we make use of a family of binary codes which we call (n, θ)-exhaustive codes. We then describe a procedure which, given any (n, V−1)-exhaustive code, constructs a test that detects all singleV-coupling faults, forn≥V>2. Following this approach, optimal (n,1)- and (n, 2)-exhaustive codes are used to construct S2CTEST and S3CTEST, which are efficient tests of length 10n and 4nlog2 n+18n that detect all single 2- and 3-coupling faults, respectively. S3CTEST is roughly five times shorter, for current RAM capacities, than Papachristou and Sahgal's test of length 24n[log2 n]+n. Codes generated according to Tang and Chen are used similarly to construct S4CTEST and S5CTEST, which are tests of approximate length 8.6n(log2 n)1.585 and 9.6n(log2 n)2.322 that detect all single 4- and 5-coupling faults, respectively. S5CTEST has the interesting property of being able to detect all single physical neighborhood pattern-sensitive faults without requiring the mapping from logical cell addresses to physical cell locations. S5CTEST also detects the scrambled pattern-sensitive fault recently proposed by Franklin and Saluja; moreover, the new test is approximately fourteen times shorter (for 1 and 4 Mbit RAMs) than the test they describe.