A class of LDPC codes based on symplectic space over finite fields

In this paper, an exact distance spectrum formula for the largest minimum Hamming distance of finite-length binary block codes is presented. The exact formula indicates that the largest minimum distance of finite-length block codes can be fully characterized by the information spectrum of the Hamming distance between two independent and identically distributed (i.i.d.) random codewords. The distance property of finite-length block codes is then connected to the distance spectrum. A side result of this work is a new lower bound to the largest minimum distance of finite-length block codes. Numerical examinations show that the new lower bound improves the finite-length Gilbert-Varshamov lower bound and can reach the minimum distance of existing finite-length block codes.

Let $q$ be a power of 2 and $\gamma\le q$ an integer. Based on the codewords of $[q,2,q-1]$ extended Reed-Solomon (RS) code over the finite field $\mathbb {F}_q$ , we can construct a $(\gamma,q)$ -regular low-density parity-check (LDPC) code, called a full-length RS-LDPC code and denoted by $\mathcal{C}(\gamma,q)$ . In this letter, the minimum distance of these codes is investigated. For any given $q$ and $\gamma\le q$ , an upper bound on $d(\mathcal{C}(\gamma,q))$ , the minimum distance of $\mathcal{C}(\gamma,q)$ , is provided. Furthermore, we determine the values of $d(\mathcal{C}(\gamma,q))$ for $\gamma=2$ , 3, and 4, and present the closed-form expressions on the numbers of minimum-weight codewords in $\mathcal{C}(\gamma,q)$ for $\gamma=2$ and 3.

We propose a technique for improving LP decoding, based on the merging of check nodes. This technique can be applied to standard as well as generalized LDPC codes. Furthermore, we show how a recently-discovered linear-complexity LP decoder can be used to derive non-trivial lower bounds on the minimum distance of specific LDPC codes, with complexity that exhibits quadratic growth with respect to the block length. This bound can be refined using the check node merging technique. The lower bound on the minimum distance is shown to be an upper bound on the fractional distance of the code.

Constructing quantum low-density parity-check (LDPC) codes with a minimum distance that grows faster than a square root of the length has been a major challenge of the field. With this challenge in mind, we investigate constructions that come from high-dimensional expanders, in particular Ramanujan complexes. These naturally give rise to very unbalanced quantum error correcting codes that have a large $X$-distance but a much smaller $Z$-distance. However, together with a classical expander LDPC code and a tensoring method that generalizes a construction of Hastings and also the Tillich--Zémor construction of quantum codes, we obtain quantum LDPC codes whose minimum distance exceeds the square root of the code length and whose dimension comes close to a square root of the code length. When the ingredient is a 2-dimensional Ramanujan complex, or the 2-skeleton of a 3-dimensional Ramanujan complex, we obtain a quantum LDPC code of minimum distance $n^{1/2}\log^{1/2}n$. We then exploit the expansion properties of the complex to devise the first polynomial-time algorithm that decodes above the square root barrier for quantum LDPC codes. Using a 3-dimensional Ramanujan complex, we also obtain an overall quantum code of minimum distance $n^{1/2}\log n$, which sets a new record for quantum LDPC codes.

We deal with the minimum distances of q-ary cyclic codes of length q^m-1 generated by products of two distinct minimal polynomials, give a necessary and sufficient condition for the case that the minimum distance is two, show that the minimum distance is at most three if q>3, and consider also the case q=3.

Finding the minimum distance of linear codes is in general a NP-hard problem, we propose an efficient algorithm to attack this problem. The principle of this approach is to search code words locally around the all-zero code word perturbed by a level of noise magnitude, in other words the maximum of noise that can be corrected by a Soft-In decoder, anticipating that the resultant nearest non-zero code words will most likely contain the minimum Hamming weight code word, whose Hamming weight is equal to the minimum distance of the linear code. A numerous results prove that the proposed algorithm is valid for general linear codes and it is very fast comparing to all others known techniques, therefore it is a good tool for computing. Comparing to Joanna's works, we proof that our algorithm has a low complexity with a fast time of execution. For some linear RQs, QDCs and BCHs codes with unknown minimum distance, we give a good estimation (true) of the minimum distance where the length is less than 439.

Twisted permutation codes, introduced recently by the second and third authors, belong to the family of frequency permutation arrays. Like some other codes in this family, such as the repetition permutation codes, they are obtained by a repetition construction applied to a smaller code (but with a "twist" allowed). The minimum distance of a twisted permutation code is known to be at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes. These constructions are based on two infinite families of finite groups and their representations. The first is a family of $p$-groups, for an odd prime $p$, while the second family consists of the $4$-dimensional symplectic groups over a finite field of even order. In the latter construction, properties of the graph automorphism of these symplectic groups play an important role.

A simple method based on Newton's identities and their extensions is presented for determining the actual minimum distance of cyclic codes. More significantly, it is shown that this method also provides a mechanism for generating the type of syndrome matrices needed by Feng and Tzeng's (see ibid., vol.40, p.1364-1374, Sept. 1994) new procedure for decoding cyclic and BCH codes up to their actual minimum distance. Two procedures for generating such matrices are given. With these procedures, we have generated syndrome matrices having only one class of conjugate syndromes on the minor diagonal for all binary cyclic codes of length n<63 and many codes of length 63/spl les/n/spl les/99. A listing of such syndrome matrices for selected codes of length n<63 is included. An interesting connection of the method presented to the shifting technique of van Lint (1986) and Wilson is also noted.

This paper examines the problem of minimising stopping distance on split friction roads by joint control of individual wheel brakes and automated steering. A static optimisation problem is introduced that maximises braking on split friction roads without veering out-of-lane. The analysis of the optimal brake forces and counter-steering shows different properties of the optimal solution depending on the degree of split friction assymmetry between the left and right vehicle sides. The solutions are categorised into two regions: small and large split friction asymmetries. At a small split friction asymmetry, all the tyres are at their force peak in their force-slip curves and the slips are small. Therefore, the traditional ABS and a path-following controller give optimal braking and path deviation performance. Whereas at a large split friction asymmetry, it is shown that having all tyres at their force-slip peak is impossible. Instead, allowing larger slips at the low-friction tyres gives maximum braking. Surprisingly, constraining slip constraint on one low-friction tyre limits the available tyre force on the high-friction ones, ultimately leading to longer stopping distances. For braking at large splits, we propose using optimal static solutions to set a feedforward steering angle based on the split friction condition, together with a high-friction brake control to the yaw torque. This control framework is tested in vehicle simulation in the CarMaker environment. Closed-loop simulations of the proposed control framework are compared to an emulated driver response derived from measurements of a professional driver. Stopping distance gains of 6–13% are observed by automating the steering compensation. Further simulations demonstrate that the proposed control framework provides maximum braking even when one side has zero friction.

Two <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -digit sequences, called "points," of binary digits are said to be at distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> if exactly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</tex> corresponding digits are unlike in the two sequences. The construction of sets of points, called codes, in which some specified minimum distance is maintained between pairs of points is of interest in the design of self-checking systems for computing with or transmitting binary digits, the minimum distance being the minimum number of digital errors required to produce an undetected error in the system output. Previous work in the field had established general upper bounds for the number of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -digit points in codes of minimum distance d with certain properties. This paper gives new results in the field in the form of theorems which permit systematic construction of codes for given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</tex> ; for some <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, d</tex> , the codes contain the greatest possible numbers of points.

A characterization of MDS codes that have an error correcting pair

When vehicle speed increases, stable and powerful brake system is required to ensure the safety of the vehicle and passengers. For high speed braking of vehicle, contactless eddy current brake (ECB) is developed and applied to antilock brake system (ABS). An adaptive sliding mode controller for ABS using eddy current is presented with a sliding mode observer (SMO). Using vehicle acceleration, wheel speeds, and braking torque, the vehicle velocity estimated by SMO. To control the system based on wheel slip, half-car model is developed by considering uncertainty of the road slope and aerodynamic drag force. According to the controller, the vehicle ABS model is built with MATLAB Simulink. The response of controller is compared in terms of braking time and stopping distance. The results indicate proposed controller algorithm achieves minimum braking time and minimum stopping distance on selected roads.

Finding the minimum distance of linear codes is one of the main problems in coding theory. The importance of the minimum distance comes from its error-correcting and error-detecting capability of the handled codes. It was proven that this problem is an NP-hard that is the solution of this problem can be guessed and verified in polynomial time but no particular rule is followed to make the guess and some meta-heuristic approaches in the literature have been used to solve this problem. In this paper, swarm-based optimization techniques, bat and firefly, are applied to the minimum distance problem by integrating the algebraic operator to the handled algorithms.

We obtain a characterization on self-orthogonality for a given binary linear code in terms of the number of column vectors in its generator matrix, which extends the result of Bouyukliev et al. (2006). As an application, we give an algorithmic method to embed a given binary k-dimensional linear code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ( k = 3,4) into a self-orthogonal code of the shortest length which has the same dimension k and minimum distance d' ≥ d( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> ). For k > 4, we suggest a recursive method to embed a k-dimensional linear code to a self-orthogonal code. We also give new explicit formulas for the minimum distances of optimal self-orthogonal codes for any length n with dimension 4 and any length n \not ≡ 6, 13,14,21,22,28,29 (mod 31) with dimension 5.

An attractive feature of BCH codes is that one can infer valuable information from their design parameters (length, size of the finite field, and designed distance), such as bounds on the minimum distance and dimension of the code. In this paper, it is shown that one can also deduce from the design parameters whether or not a primitive, narrow-sense BCH contains its Euclidean or Hermitian dual code. This information is invaluable in the construction of quantum BCH codes. A new proof is provided for the dimension of BCH codes with small designed distance, and simple bounds on the minimum distance of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families of primitive quantum BCH codes as a function of their design parameters