Constructing error-correcting binary codes using transitive permutation groups

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In thispaper, a method to conduct a computer search for binary 2-generator QC codes is presented, anda large number of good 2-generator QC codes have been obtained. 5 new binary QC codes thatimprove the lower bounds on minimum distance are presented. Furthermore, with new 2-generatorQC codes and Construction X, 2 new improved binary linear codes are obtained. With the standardconstruction techniques, another 16 new binary linear codes that improve the lower bound on theminimum distance have also been obtained.

A coordinate of a binary code of size M is said to be balanced if the number of zero and ones in the coordinate is either \(\lfloor M/2\rfloor \) or \(\lceil M/2 \rceil \) (that is, exactly \(M/2\) for even M). Since good codes (of various types) tend to be balanced in all coordinates, various conjectures have been made regarding the existence of such codes. It is here shown that there are parameters for which there are no optimal binary error-correcting codes with a balanced coordinate. This is proved by the code attaining \(A(17,8) = 36\), which is shown to be unique here; \(A(n,d)\) denotes the maximum size of a binary code of length n and minimum distance d. It is further shown that \(A(18,8) \le 68\).

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary $2$-generator QC codes is presented, and a large number of good $2$-generator QC codes have been obtained. $5$ new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new $2$-generator QC codes and Construction X, $2$ new improved binary linear codes are obtained. With the standard construction techniques, another $16$ new binary linear codes that improve the lower bound on the minimum distance have also been obtained.

An [$$n, k, d$$n,k,d] code is a binary linear code of block length $$n$$n, dimension $$k$$k and minimum Hamming distance $$d$$d. Since the minimum distance determines the error detection or correction capability, it is desired that $$d$$d is as large as possible for the given block length $$n$$n and dimension $$k$$k. One of the most fundamental problems in coding theory is to construct codes with best possible minimum distances. This problem is very difficult in both theory and practice. During the last decades, it has proved that the class of quasi-cyclic (QC) codes contain many such codes. In this paper, augmentation of binary QC codes is studied. A new augmentation algorithm is presented, and 10 new $$h$$h-generator QC codes that are better than previously known code have been constructed. Furthermore, Construction X has been applied to obtain another 18 new improved binary linear codes. With the standard construction techniques, a total of 124 new binary linear codes that improve the lower bound on the minimum distance have been obtained.

The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that $$64 \le A(18,8) \le 68$$ and $$128 \le A(19,8) \le 131$$ . In the current computer-aided study, it is shown that $$A(18,8)=64$$ and $$A(19,8)=128$$ , so an optimal code is obtained even after shortening the extended binary Golay code six times.

A binary code C ⊆ F2n with minimum distance at least d and codewords of Hamming weight w is called an (n,d,w) constant weight code. The maximum size of an (n,d,w) constant weight code is denoted by A(n,d,w), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n,d,w) constant weight codes are here classified up to equivalence for d=4, n ≤ 12; d=6, n ≤ 14; d=8, n ≤ 17; d=10, n ≤ 20 (with one exception); d=12, n ≤ 23; d=14, n ≤ 26; d=16, n ≤ 28; and d=18, n ≤ 28. Moreover, several new upper bounds on A(n,d,w) are obtained, leading among other things to the exact values A(12,4,5)=80, A(15,6,7)=69, A(18,8,7)=33, A(19,8,7)=52, A(19,8,8)=78, and A(20,8,8)=130 . Since A(15,6,6)=70, this gives the first known example of parameters for which A(n,d,w-1) > A(n,d,w) with w ≤ n/2. A scheme based on double counting is developed for validating the classification results.

For a given binary BCH code Cn of length n = 2 s - 1 generated by a polynomial of degree r there is no binary BCH code of length (n + 1)n generated by a generalized polynomial of degree 2r. However, it does exist a binary cyclic code C (n+1)n of length (n + 1)n such that the binary BCH code Cn is embedded in C (n+1)n . Accordingly a high code rate is attained through a binary cyclic code C (n+1)n for a binary BCH code Cn . Furthermore, an algorithm proposed facilitates in a decoding of a binary BCH code Cn through the decoding of a binary cyclic code C (n+1)n , while the codes Cn and C (n+1)n have the same minimum hamming distance.

We study certain projections of binary linear codes onto larger fields. These projections include the well-known projection of the extended Golay [24,12,8] code onto the hexacode over $\mbox{GF}(4)$ and the projection of the Reed--Muller code R(2,5) onto the unique self-dual [8,4,4] code over $\mbox{GF}(4)$. We give a characterization of these projections, and we construct several binary linear codes which have best known optimal parameters, for instance, [20,11,5], [40,22,8], [48,21,12], and [72,31,16]. We also relate the automorphism group of a quaternary code to that of the corresponding binary code.

Constacyclic codes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="italic">uF</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math>

Binary code negation and reversal are known to leave the autocorrelation sequence unchanged. However, for some code lengths, there exist code pairs for which the mechanism behind shared autocorrelation is not so simple; these are called shared-autocorrelation (or shared-ACS) binary code pairs. They are useful for, among other things, constructing binary and quad-phase Golay pairs of longer lengths. We define a symparity vector for any given binary code. For every shared-ACS binary code pairs up to length 39, the two code have the same symparity, or coupled symparity. Conditions are identified for which shared-ACS binary code pairs have coupled symparity. The coupled-symparity condition is used to develop an algorithm to find shared-ACS binary code pairs, with reduced complexity relative to more brute-force exhaustive searches. Although the algorithm assumes extra structure in the shared-ACS code pairs, it rediscovered all the sets of binary shared-ACS code pairs for lengths up to 20. It is then run for lengths 40 to 44, returning all the shared-ACS binary code pairs for each length. In particular, the algorithm results establish that there exist shared-ACS code pairs for the prime lengths 41 and 43.

General results on automorphisms of self-dual binary codes are given. These results are applied to the study of extremal self-dual doubly even binary codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">48</tex> . The main theorem proved is that an extremal self-dual doubly even code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">48</tex> with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code. Interesting constructions of the binary extended Golay code as well as a conjecture about a possible connection between an extremal self-dual doubly even code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">72</tex> and an extremal quaternary code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">24</tex> arc yielded by techniques used in the proof.

A triply even code is a binary linear code in which the weight of every codeword is divisible by 8. We show how two doubly even codes of lengths m_1 and m_2 can be combined to make a triply even code of length m_1+m_2, and then prove that every maximal triply even code of length 48 can be obtained by combining two doubly even codes of length 24 in a certain way. Using this result, we show that there are exactly 10 maximal triply even codes of length 48 up to equivalence.

Let G be a transitive permutation group on a set Ω such that, for ω∈Ω, the stabiliser Gω induces on each of its orbits in Ω∖{ω} a primitive permutation group (possibly of degree 1). Let N be the normal closure of Gω in G. Then (Theorem 1) either N factorises as N=GωGδ for some ω, δ∈Ω, or all unfaithful Gω-orbits, if any exist, are infinite. This result generalises a theorem of I. M. Isaacs which deals with the case where there is a finite upper bound on the lengths of the Gω-orbits. Several further results are proved about the structure of G as a permutation group, focussing in particular on the nature of certain G-invariant partitions of Ω. 1991 Mathematics Subject Classification 20B07, 20B05.

On graph-restrictive permutation groups

Asymmetric cryptographic primitives are essential to enable secure communications in public networks or public mediums. Such primitives can be deployed as software libraries or hardware co-processors, the latter being more commonly employed in systems on chip (SoC) scenarios, embedded devices, or application-specific servers. Unfortunately, the most commonly available solutions, based on RSA or elliptic curve cryptography (ECC), are highly processing intensive due to the underlying extended-precision modular arithmetic. Consequently, they are not available on highly constrained platforms. Aiming to tackle this issue, we here investigate an alternative asymmetric encryption scheme that relies on lightweight arithmetic: McEliece. This scheme is especially appealing because, being based on error correction codes, it displays a simpler arithmetic and leads to better performance when compared to RSA or ECC. To evaluate the implementation of this scheme in hardware, we propose and analyze a flexible architecture whose security level and time versus area usage characteristics can be reconfigured as desired. The proposed architecture is suitable to all usual security levels, ranging from 80 to 256 bits. It is also very efficient, being able to perform data decryption with binary Goppa codes in 56µs with 3,402 slices on a Xilinx Spartan-3AN FPGA, whereas the best-known result in the literature for the same FPGA is 115µs with 7,331 slices. Alternatively, the architecture can operate with quasi-dyadic Goppa (QD-Goppa) codes, which involves smaller keys than traditional binary Goppa codes. In the latter case, for an 80-bit security level, the decryption operation can take from 1.1ms with 1,129 slices to 68µs with 8,268 sices. By choosing a more hardware-friendly decoding algorithm, focusing hardware resources on most bottleneck operations and sharing hardware resource for two different algorithms, better results than the those in the literature were obtained.