Linear binary codes arising from finite groups

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1 Linear binary codes arising from finite groups Yannick Saouter, Member, IEEE Institut Telecom - Telecom Bretagne, Technopôle Brest-Iroise - CS Brest Cedex, France Yannick.Saouter@telecom-bretagne.eu 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. INTRODUCTION 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 Higman- Sims 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 J 1 and J 2. 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. CODES FROM PERMUTATION GROUPS Definition 2.1: Let σ 1, σ 2,..., σ n be permutations operating on the ordonned set {1, 2,..., N}. Then the set G containing σ 1, σ 2,..., σ n 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 Stab G (i). Definition 2.3: Let G be a finite permutation group acting on {1, 2,...N}, and let α = [α 1, α 2,..., α k ] 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 Orbit G (α) = {[σ(α 1 ), σ(α 2 ),..., σ(α k )] 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 α, the set Orbit G ([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 P ij = 0 or 1 and P ij = 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. MAJORITY 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, λ) near-configuration is a system of b sets and v elements whose b v binary incidence

2 matrix is such that: - Every row has exactly k 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, 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 (c 1,..., c v ) be a codeword of C. Let i, with 1 i v, be an arbitrary index of a codeword of C. We call H 0 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 c i (r estimators from H 0 and 1 estimator from received value). We dissymetrize the estimators: while estimators from H 0 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 H 0, a symbol c j with j 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 c i is not in error. Then the t errors concern only symbols c j with j i. From this we deduce W λt. Second, suppose that the received value for c i 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 c i 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. SOME 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 Higman- Sims group. This group, that we note HS, is a simple group of order 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 M 22. 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 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, J 1 and J 2. 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 M 11, M 12, M 22, M 23, M 24, Janko groups J 1, J 2, J 3, J 4, Conway groups Co 1, Co 2, Co 3, Fischer groups F i 22, F i 23, F i 24, 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 O 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 points. The largest permutation description that was effectively computed concerns B (for baby-monster ) and involves about 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

3 Group Permutation Stabilizer Orbits Orbit Code t ML size size number M M 9 : 2 (1, 18, 36) 2 (55, 45, 3) 1 M S 5 (1, 15, 20, 30) 2 (66, 44, 8) 2 M S 5 (1, 15, 20, 30) 3 (66, 56, 3) 1 M A (1, 20, 45) 2 (66, 56, 3) 1 M S 5 (1, 10 2, 15, 30 2, 60 3, 120) 10 (396, 352, 4) 1 M : S 3 (6, 16, 24, 32, 48 2, 96 3 ) 7 (495, 441, 5) 2 M : A 6 (1, 16, 60) 2 (77, 57, 5) 2 M : S 5 (1, 30, 40, 160) 3 (231, 211, 3) 1 M : L 3 (2) (1, 7, 42, 112, 168) 5 (330, 310, 4) 1 M A 6.2 (1, 30, 45, 180, 360) 5 (616, 596, 4) 1 M L 2 (11) (1, 55 2, 66, 165, 330) 4 (672, 560, 8) 3 M L 3 (4) : 2 (1, 42, 210) 2 (253, 231, 3) 1 M : A 7 (1, 112, 140) 2 (253, 231, 3) 1 M A 8 (1, 15, 210, 280) 4 (506, 484, 4) 1 M M 11 (1, 165, 330, 792) 4 (1288, 1266, 3) 1 M M 22 : 2 (1, 44, 231) 2 (276, 254, 3) 1 M : A 8 (1, 30, 280, 448) 4 (759, 737, 3) 1 M M 12 : 2 (1, 495, 792) 3 (1288, 1266, 4) 1 M : 3.S 6 (1, 90, 240, 1440) 3 (1771, 1661, 7) 3 J A 6.2 (1, 36, 108, 135) 3 (280, 266, 4) 1 Co U 6 (2) : 2 (1, 891, 1408) 3 (2300, 2278, 3) 1 HS 100 M 22 (1, 22, 77) 2 (100, 78, 6) 2 HS 1100 L 3 (4) : 2 (1, 42, 280, 672) 4 (1100,1080,3) 1 McL 275 U 4 (3) (1, 112, 162) 2 (275, 253, 5) 1 McL 2025 M 22 (1, 330, 462, 1232) 4 (2025, 2003, 3) 1 Ru 4060 T.2 (1, 1755, 2304) 3 (4060, 4032, 3) 1 T [2 8 ].5.4 (1, 10, 80, 640, 1024) 5 (1755, 1729, 3) 1 TABLE I INTERESTING MAJORITY LOGIC DECODABLE CODES OBTAINED FROM SPORADIC SIMPLE GROUPS. at most points. This criterion immediately discards the following groups: J 4 (more than points), Co 1 (98280 points), F i 23 (31671 points), F i 24 ( points), B, M, HN (more than 10 6 points), T h (more than points), O N ( points) and Ly (more than 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 Co 3. 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 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. CODES 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 S N, is the set of all bijections from [1..N] to itself. The elements σ of S N are called permutations. Definition 5.2: Let N be a positive integer, and σ S N. The signature of σ is defined by the following equation: s(σ) = 1 i<j N σ(j) σ(i) j i The signature of σ is equal either to 1 or to 1. The alternating group of order N, denoted A N, is the set of permutations σ S N whose signature is equal to 1. The entire set of subgroups of A N is not known in general. Classification of some large maximal subgroups has been

4 Group Permutation Stabilizer Orbits Graph size size M M 9 : 2 (1, 18, 36) SRG(55, 18, 9, 4) M M 9 : 2 (1, 18, 36) SRG(55, 36, 21, 28) M S 5 (1, 15, 20, 30) SRG(66, 20, 10, 4) M A (1, 20, 45) SRG(66, 20, 10, 4) M A (1, 20, 45) SRG(66, 45, 28, 36) M : A 6 (1, 16, 60) SRG(77, 16, 0, 4) M : A 6 (1, 16, 60) SRG(77, 60, 47, 45) M A 7 (1, 70, 105) SRG(176, 70, 18, 34) M A 7 (1, 70, 105) SRG(176, 105, 68, 54) M : S 5 (1, 30, 40, 160) SRG(231, 30, 9, 3) M : S 5 (1, 30, 40, 160) SRG(231, 40, 20, 4) M L 3 (4) : 2 2 (1, 42, 210) SRG(253, 42, 21, 4) M L 3 (4) : 2 2 (1, 42, 210) SRG(253, 210, 171, 190) M : A 7 (1, 112, 140) SRG(253, 112, 36, 60) M : A 7 (1, 112, 140) SRG(253, 140, 87, 65) M M 22 : 2 (1, 44, 231) SRG(276, 44, 22, 4) M M 22 : 2 (1, 44, 231) SRG(276, 231, 190, 210) J U 3 (3) (1, 36, 63) SRG(100, 36, 14, 12) J U 3 (3) (1, 36, 63) SRG(100, 63, 38, 42) J A 6.2 (1, 36, 108, 135) SRG(280, 36, 8, 4) J A 6.2 (1, 36, 108, 135) SRG(280, 135, 70, 60) Co U 6 (2) : 2 (1, 891, 1408) SRG(2300, 891, 378, 324) Co U 6 (2) : 2 (1, 891, 1408) SRG(2300, 1408, 840, 896) F i U 6 (2) (1, 693, 2816) SRG(3510, 693, 180, 126) F i U 6 (2) (1, 693, 2816) SRG(3510, 2816, 2248, 2304) HS 100 M 22 (1, 22, 77) SRG(100, 22, 0, 6) HS 100 M 22 (1, 22, 77) SRG(100, 77, 60, 56) McL 275 U 4 (3) (1, 112, 162) SRG(275, 112, 30, 56) McL 275 U 4 (3) (1, 112, 162) SRG(275, 162, 105, 81) Suz 1782 G 2 (4) (1, 416, 1365) SRG(1782, 416, 100, 96) Suz 1782 G 2 (4) (1, 416, 1365) SRG(1782, 1365, 1044, 1050) Ru 4060 T.2 (1, 1755, 2304) SRG(4060, 1755, 730, 780) Ru 4060 T.2 (1, 1755, 2304) SRG(4060, 2304, 1328, 1280) T 1600 L 3 (3) : 2 (1, 312, 351, 936) SRG(1600, 351, 94, 72) TABLE II STRONGLY REGULAR GRAPHS OBTAINED FROM SPORADIC SIMPLE GROUPS. made by Bannai [7]. In fact, here we will focus on the case of the second largest one of this classification: Theorem 5.3: Let N be a positive integer greater than 2. Then S N 2 is a maximal subgroup of A N. Those subgroups are chosen since they give a rank-3 permutation description of A N and derived error correcting codes have a modest length. This length is equal to the size of the permutation which in turn is equal to the index of the subgroup in the whole group. In this case, this value is equal to N(N 1) 2. Computations were done for all alternating groups acting on N points for 4 N 29. All cases provide classical strongly regular graphs. As for the error correcting codes, data are collected in table III. VI. CODES ARISING FROM LINEAR GROUPS In this section, we focus on another generic family of simple groups: the projective special linear groups. We begin by their definition: Definition 6.1: Let n be an integer greater or equal to 2 and let q be a prime power. Then the linear group GL(n, q) of order n over GF (q) is the group formed by n n matrices with coefficients in GF (q) such that their determinant is equal to 1 or 1. The special linear group SL(n, q) of order n over GF (q) is the subgroup of GL(n, q) formed by matrices whose determinant is equal to 1. Finally the projective special linear group P SL(n, q) (denoted also L n (q)) of order n over GF (q) is the quotient group of SL(n, q) by its subgroup formed of scalar matrices (i.e. diagonal matrices with a unique diagonal coefficient). The order of this group is equal to q n(n 1)/2 GCD(n,q 1) n 1 i=1 (qi+1 1). Not much is known in general about maximal subgroups of P SL(2, p). However, checking the Atlas of Finite Groups [4], we can notice that the dihedral groups D p 1 and D p+1 often belong to this set. We remind the reader that the dihedral group D n might be defined, for n integer, as the abstract group D n = {r, f r n, f 2, (rf) 2 }. The order of this group is 2n. Our search for codes amounts then to finding elements r and f inside L 2 (p) checking the relations in order to determine a copy of the eventual subgroup. Then, with this copy, we can

5 Group Code t ML Group Code t ML A 5 (10,6,3) 1 A 18 (153,137,3) 1 A 6 (15,11,3) 1 A 19 (171,153,3) 1 A 7 (21,15,3) 1 A 20 (190,172,3) 1 A 8 (28,22,3) 1 A 21 (210,190,3) 1 A 9 (36,28,3) 1 A 22 (231,211,3) 1 A 10 (45,37,3) 1 A 23 (253,231,3) 1 A 11 (55,45,3) 1 A 24 (276,254,3) 1 A 12 (66,56,3) 1 A 25 (300,276,3) 1 A 13 (78,66,3) 1 A 26 (325,301,3) 1 A 14 (91,79,3) 1 A 27 (351,325,3) 1 A 15 (105,91,3) 1 A 28 (378,352,3) 1 A 16 (120,106,3) 1 A 29 (406,378,3) 1 A 17 (136,120,3) 1 A 30 (435,407,3) 1 TABLE III ERROR CORRECTING CODES OBTAINED FROM GROUPS A N, 4 N 29. Bit Error Rate 1e+00 1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1e-08 Fig. 1. LDPC(136,64) LDPC(276,120) LDPC(406,196) Eb/N0 (db) Performances of LDPC codes from finite groups 5 Group Stab. Code Group Stab. Code L 2 (5) D 3 (10,4,4) L 2 (17) D 9 (136,64, 9) L 2 (5) D 2 (15,7,5) L 2 (17) D 8 (153,54, 9) L 2 (7) D 3 (28,8,8) L 2 (19) D 9 (190,80, 9) L 2 (7) D 4 (21,7,3) L 2 (19) D 10 (171,56, 5) L 2 (11) D 5 (66,24,12) L 2 (23) D 11 (276,120, 13) L 2 (11) D 6 (55,21,9) L 2 (23) D 12 (253,67, 7) L 2 (13) D 7 (78,36,12) L 2 (29) D 15 (406,196, 17) L 2 (13) D 6 (91,29,13) L 2 (29) D 14 (435,90, 9) TABLE IV LDPC CODES OBTAINED FROM GROUPS L 2 (p), 5 p 29. compute the permutation representation of L 2 (p) as well as the incidence matrix of codes. In our computations, we test prime numbers p from 5 to 29, for both possible subgroups. Contrary to the case of alternating groups, the permutations we obtain have many orbits. The generated codes, although being decodable by majority logic techniques, have quite low minimum distance compared to the optimum for same code. However, we obtained many LDPC codes whose minimum distance is quite large for LDPC codes. Those codes are collected in table IV. LDPC codes arising from projective special linear groups are obviously candidates to soft decoding. In our applications, we used classical belief propagation [8] to decode them. The maximum number of iterations was set to 50 and decoding computations were performed in arbitrary precision. Figure 1 illustrates performance of some of these codes with encoding rate near 1/2. codes. REFERENCES [1] V.D. Tonchev. Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs. IEEE Trans. Inf. Theory, 43(3): , May [2] J.D. Key and J. Moori. Codes, designes and graphs from the Janko groups J 1 and J 2. Journal Combin. Math. and Combin. Comput., pages , [3] L.D. Rudolph. A class of majority logic decodable codes. IEEE Trans. on Inf. Theory, pages , Apr [4] J.H. Conway, R.T. Curtis, S.P. Norton ans R.A. Parker, and R.A. Wilson. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Clarendon Press, Oxford, See also: [5] The GAP Group. GAP Groups, Algorithms, and Programming, Version , [6] A.E. Brouwer. Parameters of Strongly Regular Graphs. aeb/graphs/srg/srgtab.html. [7] E. Bannai. Maximal subgroups of low rank of finite symmetric and alternating groups. J. Fac. Sci. Univ. Tokyo, pages , [8] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, 2nd edition, VII. CONCLUSION In this article, we have shown that the study of finite groups can be used to design new majority logic decodable codes. Some of them turn out to be LDPC codes and can be decoded by belief propagation. Some of these codes are also linked to strongly regular graphs. Further research could focus on strongly regular graphs not derived from finite groups to determine whether they can also be used to build interesting