Mathematica Balkanica. Binary Self-Dual Codes Having an Automorphism of Order p 2. Stefka Bouyuklieva 1, Radka Ruseva 2, Nikolay Yankov 3

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1 Mathematica Balkanica New Series Vol. 9, 2005, Fasc. -2 Binary Self-Dual Codes Having an Automorphism of Order p 2 Stefka Bouyuklieva, Radka Ruseva 2, Nikolay Yankov 3 We describe a method for constructing binary self-dual codes having an automorphism of order p 2 for an odd prime p. Using this method, we classify the self-dual [50, 25, 0] and [52, 26, 0] codes with an automorphism of order 25. Moreover, we prove that these codes are the only optimal self-dual codes of length less than 00 having an automorphism of order 25. We use the same method to construct the extremal self-dual codes of lengths 36, 38, and 40, having an automorphism of order 9. AMS Subj. Classification: 05A05 Key Words: self-dual codes, automorphisms of codes.introduction Let C be a binary self-dual code of length n and σ be an automorphism of C of order p 2 for the odd prime p. Without loss of generality we can assume that: () σ = Ω...Ω c Ω c+...ω c+t Ω c+t+...ω c+t+f where Ω i = ((i )p 2 +,...,ip 2 ),i =,...,c are the cycles of length p 2, Ω c+i =(cp 2 +(i )p +,...,cp 2 + ip),i =,...,t are the cycles of length p, andω c+t+i =(cp 2 + tp + i),i =,...,f are the fixed points. Obviously, cp 2 + tp + f = n. Let F σ (C) = {v C : vσ = v} and E σ (C)={v C : wt(v Ω i ) 0(mod 2),i =,...,c+ t + f}, wherev Ω i is the restriction of v on Ω i. Then the following Lemma holds Lemma.. [4] ThecodeCisadirectsumofthesubcodesF σ (C) and E σ (C). Partially supported by Grant No N22/ with Konstantin Preslavsky University, Shoumen

2 26 S. Bouyuklieva, R. Ruseva, N. Yankov Clearly v F σ (C) iff v C and v is constant on each cycle. Let π : F σ (C) F c+t+f 2 be the projection map where if v F σ (C), (vπ) i = v j for some j Ω i,i=, 2,...,c+ t + f. Theorem.2. If C is a binary self-dual code having an automorphism σ of type () then π(f σ (C)) is a binary self-dual code of length c + t + f. The proof is similar to the proof of Lemma in [3]. Denote by E σ (C) the code E σ (C) with the last f coordinates deleted. So E σ (C) is self-orthogonal binary code of length cp 2 + tp and dim(e σ (C) )= 2 (cp2 + tp + f c t f) = 2 (c(p2 ) + t(p )) = 2 (p )(c(p +)+t). For v in E σ (C) we identify v Ω i =(v 0,v,,v p 2 ) with the polynomial v 0 + v x + + v p 2 x p2 from P for i =,...,c,andv Ω i =(v 0,v,,v p ) with the polynomial v 0 + v x + + v p x p from P 2 for i = c +,...,c+ t, where P and P 2 are the sets of even-weight polynomials in F 2 [x]/(x p2 ) and F 2 [x]/(x p ), respectively. Thus we obtain the map φ : E σ (C) P c P 2 t. It is easy to see that + x + x x p2 = Q p (x)q p 2(x), where Q p and Q p 2 arecyclotomicpolynomials. If2isamultiplicativerootmodulop 2 these two polynomials are irreducible. Then P = I I 2,whereI and I 2 are the cyclic codes with parity check polynomials Q p (x) andq p 2(x), respectively. It is known that I = GF (2 p )andi 2 = GF (2 p 2 p ) (for details see [5]). To continue our investigations, we need to prove some properties of the binary linear codes of length cp 2 having an automorphism τ of order p 2 with c independent p 2 -cycles. Let A be such a code and M j = {u E τ (A) u i I j },i=, 2. Then M j is a linear space over I j,j =, 2, and the following Lemma holds: Lemma.3. If A is a binary linear code of length cp 2 having an automorphism of order p 2 with c independent p 2 -cycles then M = φ(e τ (A)) = M M 2 and (p )dim I M +(p 2 p)dim I2 M 2 = dime τ (A) Proof. Let e I and e 2 I 2 are the identities of the two fields. Then e and e 2 are primitive idempotents of the ring P and so e (x)e 2 (x) =0and e(x) =e (x)+e 2 (x) wheree(x) =x + x x p2 is the identity of P (see [5]). It is easy to see that M = Me (x)+me 2 (x) andme j (x) =M j,j =, 2. Since I I 2 = {0} then M M 2 = {0}. HenceM = M M 2. We can consider I and I 2 as binary cyclic codes with dimensions p and p(p ) respectively. Hence dime τ (A) =dimφ (M )+dimφ (M 2 )=(p )dimm + p(p )dimm 2 Obviously, σ p is an automorphism of C of type p (cp, tp+f). Let F 0 (C) and E 0 (C) are the corresponding codes. Then E 0 (C) E σ (C) andf σ (C) F 0 (C).

3 Binary Self-Dual Codes We can prove the following inequalities: Theorem.4. () p 2 c P (p )pc/2 i=0 d d e 2 i (2) If tp + f>cpthe following inequality holds tp + f (tp+f cp)/2 X i=0 d d 2 i e (3) If f>c+ t the following inequality holds f (f c t)/2 X i=0» d 2 i ¼ The proof is similar to the proof of Theorem in [6]. The main tool is Griesmer bound [5]. 2. Self-dual codes with an automorphism of order 25 We consider binary self-dual code of length less than 00 having an automorphism σ of type 25-(c,t,f), i.e. σ have c independent cycles of length 25, t independent cycles of length 5 and f fixed points. Then σ 5 is an automorphism of type 5 (5c, 5t+f) and since c must be even we have, that n 50. Using the inequalities (), (2) and (3) we reject all the parameters t and f = n 50 5t except the following four cases: [50, 25, 0] codes with c =2,t= f =0 A self-dual code of length 50 has an automorphism of order 25 with two cycles iff it is pure double circulant code. All extremal double circulant self-dual codes of length up to 62 are classified in [2]. There exist exactly two inequivalent pure double circulant self-dual [50,25,0] codes. Both codes have automorphism groups of order 50. [52, 26, 0] codes with c =2,t=0,f =2 A self-dual code of length 52 has an automorphism of order 25 with two cycles iff it is bordered double circulant code. There exist exactly two inequivalent bordered double circulant self-dual [52,26,0] codes. Both codes have automorphism groups of order 50. [56, 28, 2] codes with c =2,t= f = Let C be a self-dual [56,28,2] self-dual code with an automorphism σ with c =2,t = f =. LetA = { v C : v =(v,v 2,...,v 50, 0, 0, 0, 0, 0, 0)}

4 28 S. Bouyuklieva, R. Ruseva, N. Yankov and A be the code A with the last six coordinates deleted. Obviously, A is a self-orthogonal code of length 50 and dimension 22 having an automorphism τ of order 25 with two cycles of the same length. According Lemma.3, dime τ (A )=4dimM +20dimM 2 and therefore 4 divides dime τ (A ).Butthisisimpossibleas(00...0) and (...) are the only vectors of even weight in F 50 2 which τ preserves. [60, 30, 2] codes with c =2,t=2,f =0 Let C be a self-dual [60,30,2] self-dual code with an automorphism σ with c = t =2,f =0. Wehavethatπ(F (C)) is a [4, 2, 2] binary self-dual code. If (00) π(f (C)) then π (00) is a vector of weight 0 in C, but the minimum weight of C is 2, so π(f (C)) = {(0000), (00), (00), ()}. Let A = { v C : v =(v,v 2,...,v 50, 0,...,0)} and A be the code A with the last ten coordinates deleted. Obviously, A is a self-orthogonal code of length 50 and dimension 20 having an automorphism of order 25 with two cycles of the same length. According Lemma.3, dime τ (A )=4dimM + 20dimM 2 and we have the only possibility dimm =0, dimm 2 =. Let (e 2 g(x)) be a generator matrix of M 2. It is easy to see that C has a generator matrix in the form G = φ (e 2, g(x), 00000, 00000) φ (xe 2, xg(x), 00000, 00000) φ (x 9 e 2, x 9 g(x), 00000, 00000) φ (e, 0, x+ x 2 + x 3 + x 4, 00000) φ (xe, 0, +x 2 + x 3 + x 4, 00000) φ (x 2 e, 0, +x + x 3 + x 4, 00000) φ (x 3 e, 0, +x + x 2 + x 4, 00000) φ (0, e, 00000, x+ x 2 + x 3 + x 4 ) φ (0, xe, 00000, +x 2 + x 3 + x 4 ) φ (0, x 2 e, 00000, +x + x 3 + x 4 ) φ (0, x 3 e, 00000, +x + x 2 + x 4 ) where e 2 (x) =x 20 + x 5 + x 0 + x 5 and e (x) =x x e 2 (x). But the sum of the 2-th and 29-th rows is a codeword of weight 6 so in this case there are no extremal codes.

5 Binary Self-Dual Codes Theorem 2.. The two pure double circulant [50,25,0] and the two bordered double circulant [52,26,0] are the only extremal self-dual codes of length less than 00 having an automorphism of order Self-dual codes with an automorphism of order 9 Let C be an extremal self-dual code of length 36 n 40 with an automorphism σ =(, 2,..., 9)(0,,..., 8)...(28, 29,..., 36) with f fixed points. The polynomial h(x) =x 2 +x+ is irreducible over F 2. Hence P = I I 2 where I = {0,x s e,s =0,, 2} is a field with 4 elements and with identity e = x 8 + x 7 + x 5 + x 4 + x 2 + x, andi 2 is a fieldwith2 6 elements and identity e 2 = x 6 + x 3. The element α =(x +)e 2 is a primitive element in I 2. We consider the element t = α 9 = x 2 + x 4 + x 5 + x 7 of multiplicative order 7 in I 2. Then I 2 = {0,x s t k,for0 s 8and0 k 6}. Soφ(Eσ)=M M 2 where M and M 2 are linear [4, 2] codes over the fields I and I 2, respectively. Because of the minimal distance 8 in C and the orthogonal condition we obtain that M is a [4,2,2] code and M 2 is a [4,2,3] MDS code. We look for the generator matrix à of φ(e! σ)intheform geni2 M G = 2. gen I M Ã! e2 t t Up to equivalence we can take gen I2 M 2 = 3 e 2 t 3. t The permutations (23) and (24) are automorphisms of the code gen I2 M 2. Using à them we obtain four! possibilities à L i,i=,...4forgen! I M : e e L = e xe,l e e 2 =, e e Ã! Ã! e xe L 3 = e xe and L e xe 4 = e x 2 e Denote by G i the generator matrix of φ(e σ), corresponding to L i for i =...4. We fix the generator matrix of φ(e σ) and consider all possibilities for C π We look for the generator matrix of the code C in the form genc = gen(φ(e σ)) genf σ (C) f fixed points.

6 30 S. Bouyuklieva, R. Ruseva, N. Yankov The automorphism σ partitions the set of vectors of the code C into orbits of length, 3 or 9. Any vector of F σ (C) is in an orbit of length. Denote by A i and B i the coefficients in the weight enumerators of the codes C and F σ (C), respectively. Hence A i B i (mod 3). For n=36, f = t =0andC π is a [4,2] self-dual code. There exist 3 inequivalent [36,8,8] self-dual codes with an automorphism of order 9: when C π = {(0000), (00), (00), ()} and gen(φ(eσ)) = G 2 or G 3, when C π = {(0000), (00), (00), ()} for gen(φ(eσ)) = G 4. For n=38, f =2,t =0andC π is a [6,3,2] self-dual code, equivalent to the code C2 3. There exist 3 inequivalent [38,9,8] self-dual codes with an automorphism of order 9: when C π =< (0000), (0000), (0000) > and gen(φ(eσ)) = G 4 ; when C π =< (0000), (0000), (0000) > for gen(φ(eσ)) = G 2 or G 3. The three codes have the same weight enumerator with 7 codewords of weight 8. The orders of their automorphism groups are 9, 342, and 8, respectively. When n =40,t=0,f =4,C π is a [8,4] self-dual code. Up to equivalence, two such code exist and they are C2 4 and to the extended Hamming code H 8. Since the minimal weight of C is 8, we can take a generator matrix of the code C π in the standard form (I 4 A), where I 4 is the identity matrix and A = if C π = C2 4,andA = 0 0 if C π = H Let C be a singly-even [40,20,8] code. Then its weight enumerator is W (y) = + (25 + 6β)y 8 + (664 64β)y 0 + where 0 β 0 is a parameter []. The subcode F σ (C) does not contain vectors of weight 8 and so A 8 = β 0(mod 3). Hence β (mod 3) and A (mod 3). If F σ (C) contain vectors of weight 0 its image under the map π will be of weight 2. So the code C π must be equivalent to C 4 2. We obtain four inequivalent singly-even [40,20,8] codes with an automorphism σ. The first one has an weight enumerators W for β =0andG generates the image of its subcode E σ under the map φ. Its automorphism group is of order 8432 = 2 9. The other three codes have weight enumerators W for β = and for them gen(φ(e σ)) = G 2,G 3 and G 4, respectively. The orders of their automorphism groups are 8, 36 and again 36. Using the extended Hamming code H 8, we obtain four inequivalent doubly-even [40,20,8] codes with an automorphism σ. For these codes gen(φ(e σ))

7 Binary Self-Dual Codes... 3 = G i for i =, 2, 3, 4. They have automorphism groups of orders 8432, 8, 6840, and 36. References [] J. H. C o n w a y, N. J. A. S l o a n e. A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory 36, 990, [2]M.Harada,T.A.Gulliver,H.Kaneta. Classification of extremal double circulant self-dual codes of length up to 62, Discrete Math., 88, 998, [3] W. C. H u f f m a n. Automorphisms of codes with application to extremal doubly-even codes of length 48, IEEE Trans. Inform. Theory 28, 982, [4] W. C. H u f f m a n. Decomposing and shortening codes using automorphisms, IEEE Trans. Inform. Theory, 32, 986, [5]V.Pless,W.C.Huffman. Handbook of Coding Theory, VolumeI,Elsevier, Amsterdam (998). [6] V. Y. Y o r g o v. Binary self-dual codes with an automorphism of odd order, Problems Inform.Transm., 4, 983, 3 24 (in Russian). Department of Mathematics and Informatics Received Veliko Tarnovo University Veliko Tarnovo 5000, BULGARIA stefka@uni-vt.bg 2 Faculty of Mathematics and Informatics Shumen University Shumen 972, BULGARIA radka russeva@yahoo.com 3 Faculty of Mathematics and Informatics Shumen University Shumen 972, BULGARIA jankov niki@yahoo.com