Linear binary codes arising from finite groups Yannick Saouter, Member, IEEE
Institut Telecom - Telecom Bretagne, Technopˆole Brest-Iroise - CS 83818 29238 Brest Cedex, France Email:
[email protected]
Abstract— In this paper, we describe constructions of majority logic decodable codes which stem out description of finite groups. To this end, we generalize some previous studies of Tonchev, Key and Moori on codes from finite groups. We also extend some results obtained by Rudolph for majority logic decodable codes. Finally, we describe applications of these codes in the area of soft-decoding techniques.
I. I NTRODUCTION During the last century, the classification of finite simple groups was a great task in group theory. Simple groups are defined as groups for which no proper subgroup is stable by the action of interior automorphism. The classification is now considered finished by most group theorists. This work led to the discovery of many groups belonging to infinite families as well as 26 isolated groups, that are called sporadic groups. It is also noteworthy that computers played a significant role in this classification. For instance, the first existence proof of the Lyons sporadic group was established by a computer construction of this group. It was also noted that some of these groups give rises of particular combinatorial structures such as designs or strongly regular graphs. Sometimes those structures play a central role for the group. For instance, the Higman-Sims group was initially defined as the automorphism group of a 100 vertices strongly regular graph. In [1], Tonchev shows that the HigmanSims graph can be used to define majority logic decodable codes. He also notes that some of these codes are very close to optimal codes in terms of minimal distance. In [2], Key and Moori perform an almost systematic search of combinatorial structures arising from the Janko groups J1 and J2 . They also described some codes but did not study them from the point of view of majority logic decoding. In this article, we begin by reviewing the codes of Tonchev, Key and Moori. Then we show that other sporadic groups also give remarkable codes. We then describe two generic families of majority logic decodable codes and finally we describe some applications of these codes to soft decoding. However, we first review some basics of group theory that will be necessary for the definitions of these codes. II. C ODES
FROM PERMUTATION GROUPS
Definition 2.1: Let σ1 , σ2 , ..., σn be permutations operating on the ordonned set {1, 2, ..., N }. Then the set G containing σ1 , σ2 , ..., σn and closed for the function composition ◦ is a group for the operation ◦.
Definition 2.2: Let G be a permutation group acting on {1, 2, ..., N }. Then the stabilizer for i, 1 ≤ i ≤ N , is the subgroup of G containing all permutations σ such that σ(i) = i. The stabilizer for i in G is denoted StabG (i). Definition 2.3: Let G be a finite permutation group acting on {1, 2, ...N }, and let α = [α1 , α2 , ..., αk ] be a vector of elements of {1, 2, ...N }, then the orbit of α in G is the set of the vector images of α by permutations in G, thus OrbitG (α) = {[σ(α1 ), σ(α2 ), ..., σ(αk )] | σ ∈ G}. The final theorem comes from [2] and explicits the construction of our codes: Theorem 2.4: Let G be a finite permutation group acting on {1, 2, ...N } and let α ∈ {1, 2, ...N }. Let H be the stabilizer of α in G. The set of orbits of single elements of {1, 2, ...N } in H form a partition of the set {1, 2, ...N }. The number of partitions is called rank. Then for any β in {1, 2, ...N } with β 6= α, the set OrbitG ([α, β]) forms the edge set of a regular connected graph, with G acting as an automorphism group on this graph. In some cases, the previous graph appears to be strongly regular: Definition 2.5: A (v, k, λ, µ) strongly regular graph is a regular graph with v vertices, whose valency is k and such that any two neighbour vertices of the graph have λ common neighbours while any two non-neighbour vertices of the graph have µ common neighbours. The related binary code is then built from the edge set of the graph by defining the N × N parity matrix P of the code by Pij = 0 or 1 and Pij = 1 if and only if there exists an element g ∈ G such that i = g(α) and j = g(β). Since the graph is regular, the number of non null entries in rows of P is a constant and is equal to the number of non null entries in columns of P . In fact this number is equal to the number of points in the orbit in H containing β. III. M AJORITY
LOGIC DECODABLE CODES FROM CONFIGURATIONS
In [3], Rudolph defines a combinatorial structure called configuration and shows that such objects can be used to define majority logic decodable codes. In our work, we need a generalization of this result. To this end, we will first introduce a generalisation of Rudolph’s structures, that we call nearconfiguration: Definition 3.1: An (b, v, r, k 0 , λ) near-configuration is a system of b sets and v elements whose b × v binary incidence
matrix is such that: - Every row has exactly k 0 entries equal to 1. - Every column has exactly r entries equal to 1. - The scalar product of any two columns is smaller than λ. We have then: Theorem 3.2: Let C be the binary code whose b × v parity matrix is the incidence matrix of a (b, v, r, k 0 , λ) nearconfiguration. Then the majority logic decoding procedure can be used to decode any combination of t errors, provided that 0 ≤ t < r+λ 2λ . Proof: Let (c1 , ..., cv ) be a codeword of C. Let i, with 1 ≤ i ≤ v, be an arbitrary index of a codeword of C. We call H0 the r × v submatrix of the parity matrix of C whose rows have an entry equal to 1 at index i. We have then r + 1 estimators for ci (r estimators from H0 and 1 estimator from received value). We dissymetrize the estimators: while estimators from H0 will have a contribution weight of 1 in the decision, the estimator from the received value will be assigned a weight of λ. The total weight for decision will be then r +λ. Amongst the estimators from H0 , a symbol cj with j 6= i can occur in no more than λ ones. Suppose now that the codeword contains t errors and we denote W the global weight of wrong estimations. The final decision will be correct if and only if r + λ > 2W . First, suppose that the received value for ci is not in error. Then the t errors concern only symbols cj with j 6= i. From this we deduce W ≤ λt. Second, suppose that the received value for ci is in error. Then other symbols contain at most t − 1 errors and thus have a contributing weight which is upper bounded by λ(t−1). The received value contributes for λ in the weight and thus again we have W ≤ λt. The final decision for ci will be then surely correct if (r + λ) > 2.λt. Thus if the number t of errors in the codeword is such that t < r+λ 2λ any symbol can be corrected. As we will see on the following, structures arising from finite groups by construction of theorem 2.4 will be in fact nearconfigurations. IV. S OME
CODES ARISING FROM SPORADIC GROUPS
While preceding constructions can be applied to arbitrary groups, we focus here on simple groups for which a large documentation is available [4]. In this section, we begin by examples on some sporadic groups, already studied [1], [2]. A. The Higman-Sims group In [1], a majority logic decodable code is built with HigmanSims group. This group, that we note HS, is a simple group of order 44252000. From [4], we obtain descriptions for this group: Theorem 4.1: The group HS can be described as a permutation group acting on 100 points, from the action of 2 generators on cosets of M22 . We used GAP [5] for computations. By applying theorem 2.4, this representation gives a partition of the 100 points into three orbits: one of length 1 (corresponding to α), one
of length 22 and one of length 77. If we choose β in the orbit of length 77, the incidence matrix is effectively a nearconfiguration. However, the rank of this matrix is equal to 100, and thus cannot be the parity matrix of a code. If we choose β in the orbit of length 22, we obtain a (100, 100, 22, 22, 6) configuration. This configuration, in turn gives the well-known Higman-Sims graph, which is strongly regular. The rank of the incidence matrix is 22. Thus it gives a (100, 78) code. From computations, Tonchev established that this code has a minimum distance equal to 6 which is near the optimum value of 8 for the given parameters of the code. By applying theorem 3.2, majority logic decoding can correct up to 2 errors which is exactly the power of correction of the code. Tonchev only focused on the length 100 permutation description of the HS group. However, the HS group possesses larger permutation descriptions which can again provide majority logic decodable codes. B. Janko groups The four Janko groups belong to the class of sporadic simple groups. In spite of a common denomination, they do not have relation with each other. In fact, they were discovered by the same mathematician, Z. Janko. In article [2], the authors made a computational inspection of the two first groups, J1 and J2 . Their work was essentially motivated by combinatorial considerations. They essentially focus on searching strongly regular graphs and correcting codes but do not deal with the majority logic decoding aspect. In their work, they recover some known strongly regular graphs and they generated many correcting codes with wide automorphism group. Some points of this study differ from Tonchev’s. First, the study was not limited to minimal size permutation descriptions. Second, while Tonchev relates the Higman-Sims code to the rank 3 permutation description of the HS group, Key and Moori also investigate permutation description of higher rank. C. General case of sporadic simple groups In this paragraph, we generalize the preceding works to the case of all 26 sporadic simple groups. Those groups are the following: Mathieu groups M11 , M12 , M22 , M23 , M24 , Janko groups J1 , J2 , J3 , J4 , Conway groups Co1 , Co2 , Co3 , Fischer groups F i22 , F i23 , F i024 , B and M , Higman-Sims group HS, McLaughlin group M cL, Suzuki group Suz, Held group He, Harada-Norton group HN , Thomson group T h, O’Nan group O0 N , Rudvalis group Ru, Lyons group Ly and finally Tits group T . Although we intend to make a systematic search for structures and groups, some practical limitations arise. Indeed, for instance, the largest group of this list M (for “monster”) has no known description as permutation group. The only thing which is known about this description is that it is acting on more than 9.7 × 1019 points. The largest permutation description that was effectively computed concerns B (for “baby-monster”) and involves about 1.3 × 1010 points. Since Gaussian eliminations are necessary to compute the rank of codes, large permutation groups cannot be treated. We then decided to limit ourselves to permutation description action on
Group M11 M11 M11 M12 M12 M12 M22 M22 M22 M22 M22 M23 M23 M23 M23 M24 M24 M24 M24 J2 Co2 HS HS M cL M cL Ru T
Permutation size 55 66 66 66 396 495 77 231 330 616 672 253 253 506 1288 276 759 1288 1771 280 2300 100 1100 275 2025 4060 1755
Stabilizer M9 : 2 S5 S5 A6 .22 2 × S5 21+4 : S3 2 4 : A6 2 4 : S5 23 : L3 (2) A6 .2 L2 (11) L3 (4) : 2 2 4 : A7 A8 M11 M22 : 2 2 4 : A8 M12 : 2 26 : 3.S6 3.A6 .2 U6 (2) : 2 M22 L3 (4) : 2 U4 (3) M22 T.2 2.[28 ].5.4
Orbits size (1, 18, 36) (1, 15, 20, 30) (1, 15, 20, 30) (1, 20, 45) (1, 102 , 15, 302 , 603 , 120) (6, 16, 24, 32, 482 , 963 ) (1, 16, 60) (1, 30, 40, 160) (1, 7, 42, 112, 168) (1, 30, 45, 180, 360) (1, 552 , 66, 165, 330) (1, 42, 210) (1, 112, 140) (1, 15, 210, 280) (1, 165, 330, 792) (1, 44, 231) (1, 30, 280, 448) (1, 495, 792) (1, 90, 240, 1440) (1, 36, 108, 135) (1, 891, 1408) (1, 22, 77) (1, 42, 280, 672) (1, 112, 162) (1, 330, 462, 1232) (1, 1755, 2304) (1, 10, 80, 640, 1024)
Orbit number 2 2 3 2 10 7 2 3 5 5 4 2 2 4 4 2 4 3 3 3 3 2 4 2 4 3 5
Code parameters (55, 45, 3) (66, 44, 8) (66, 56, 3) (66, 56, 3) (396, 352, 4) (495, 441, 5) (77, 57, 5) (231, 211, 3) (330, 310, 4) (616, 596, 4) (672, 560, 8) (253, 231, 3) (253, 231, 3) (506, 484, 4) (1288, 1266, 3) (276, 254, 3) (759, 737, 3) (1288, 1266, 4) (1771, 1661, ≥ 7) (280, 266, 4) (2300, 2278, 3) (100, 78, 6) (1100,1080,3) (275, 253, 5) (2025, 2003, 3) (4060, 4032, 3) (1755, 1729, 3)
tM L 1 2 1 1 1 2 2 1 1 1 3 1 1 1 1 1 1 1 3 1 1 2 1 1 1 1 1
TABLE I I NTERESTING MAJORITY LOGIC DECODABLE CODES OBTAINED FROM SPORADIC SIMPLE GROUPS .
at most 10000 points. This criterion immediately discards the following groups: J4 (more than 1.7×108 points), Co1 (98280 points), F i23 (31671 points), F i024 (306936 points), B, M , HN (more than 106 points), T h (more than 1.4 × 108 points), O0 N (112760 points) and Ly (more than 8 × 106 points). Moreover, we only focus on permutation description obtained from maximal subgroups. Amongst the remaining groups, eventual permutation descriptions of rank 2 will also have to be discarded since they cannot provide interesting structures. This condition discards Co3 . Since we are especially interested in majority logic decoding and its application to decoding, we also discard eventual codes whose majority logic decoding power of correction is too small compared to half the minimum distance of the code. Results of computations are summarized in table I. In this table, stabilizers are described using notations of [4]. We remark immediately that the vast majority of interesting codes are in fact given by Mathieu groups. During our computations, we also find strongly regular graphs. The parameters of these graphs are summarized in table II. Most of them are classical, in the sense that they arise from rank 3 orbits of corresponding groups. However, following [2], we also remark that some of them come from rank 4 orbits. We found six such graphs. The first five are recorded in Brouwer’s database of strongly regular graphs [6]. The last one, built with Tits’group, although already known, possesses few quotations
in the scientific production. V. C ODES
ARISING FROM ALTERNATING GROUPS
In the previous part, we have seen that sporadic groups can be used to generate a large number of error correcting codes, as well as strongly regular graphs. However, we have obtained only a finite number of such structures. In this part, as well as in the following one, we exhibit a generic family of constructions. This family is indexed by an integer N ≥ 5. This family comes out from the classical alternating groups. We first introduce those groups. Definition 5.1: Let N be a positive integer. Then the symmetric group of order N , denoted SN , is the set of all bijections from [1..N ] to itself. The elements σ of SN are called permutations. Definition 5.2: Let N be a positive integer, and σ ∈ SN . The signature of σ is defined by the following equation: Y σ(j) − σ(i) s(σ) = j −i 1≤i
