Twisted Tensor Product Codes

Transcription

1 Twisted Tensor Product Codes Anton Betten Department of Mathematics Colorado State University October, 2007

2 Abstract Two families of constacyclic codes with large automorphism groups will be presented. They are obtained from the twisted tensor product construction. Coincidentally, the smallest nontrivial member of the first family is the quaternary self-orthogonal code S18 discussed by MacWilliams, Odlyzko, Sloane, and Ward in 1978.

3 References Anton Betten: Twisted Tensor Product Codes, to appear in Designs, Codes, Cryptography. Antonio Cossidente, Oliver King: Twisted Tensor product group embeddings and complete partial ovoids on quadrics in PG(2 t 1, q). J. Algebra 273 (2004)

4 THEOREM 1 A) constacyclic [q 2 + 1, q 2 8, 6] q any q 3. They are cyclic if and only if q is even. B) [q 2 + 2, q 2 7, 6] q codes any q 4 even In both case, ΓAut = PΓL(2, q 2 ), and PΓ 2 L(2, q 2 ) MAut (throughout, q denotes a prime power) PAut MAut ΓAut (as in Pless / Huffman)

5 Example B) for q = 4 gives the [18, 9, 8] 4 code S 18 of MacWilliams, Odlyzko, Sloane, Ward (JCT-A 1978) Here, 2 and 3 denote ω and ω 2 in F 4, respectively. φ 2 is the Frobenius Automorphism of F 4 over F 2. The code is formally self-dual (b/c its dual is φ 2 (S 18 ), its complex conjugate, and conjugation preserves weights).

6 The Automorphism group of S 18 Its automorphism group is PΓL(2, 16) orbit structure: Q: How can PΓL(2, 16) act on a code defined over F 4? A: the twisted tensor product action (see below)

7 THEOREM 2 constacyclic [q 3 + 1, q 3 7, 5] q any q 3 ΓAut = PΓL(2, q 3 ), PΓ 3 L(2, q 3 ) MAut The codes are cyclic if and only if q is even.

8 The Construction Let V n = F n q t be an n-dimensional vector space over F q t. Consider t V n := V n V n V n (t times) Define a mapping ι t : V n t V n, x x φ t (x) φ 2 t (x) φ t 1 t (x). This induces a mapping between the corresponding projective spaces: ι t : P(V n ) P( t V n ) : P(x) P(ι t (x)).

9 Projective Codes { isometry classes of projective code } { equivalence classes of n points in projective space } Often, a [n, k] q code can be identified with a set of n points in PG(k 1, q). The columns of the generator matrix are the homogeneous coordinates of the n points. Conversely, n points in PG(k 1, q) may be considered as generating a code. The property minimum distance d corresponds to the geometric property any d 1 points are independent

10 The Construction The Veronese map ν t : PG(1, q) PG(t 1, q) P(a, b) P(a t, a t 1 b,..., b t ) ι 2 ν 3 (PG(1, q 2 )) gives a set of n = q points in PG(8, q 2 ) Since it is a twisted tensor product image, it lies in a PG(8, q) subfield subspace. This gives the codes of THEOREM 1 of length n = q For length n = q 2 + 2, add the nucleus to the image ν 3 (PG(1, q 2 )) ι 3 (PG(1, q 3 )) gives a set of n = q points in PG(7, q 2 )

11 The Twisted Tensor Product Action Let G PΓL(n, q t ) G acts on V n. G also acts on t V n, namely (v 1 v t, g) v 1 g φ t (v 2 g) φ 2 t (v 3, g) φ t 1 t (v t g) Let ρ(g) denote this action.

12 Are they BCH codes? For n = q 2 + 1, take the cyclotomic sets of 0, 1, 2 mod q 2 + 1: {0} {1, q, q 2 1, q, q 2 1} {2, 2q, 2q 2 2, 2q, 2q 2 2} 9 roots, in order: 2q, q, 2, 1, 0, 1, 2, }{{} consecutive set q, 2q, This yields a [q 2 + 1, q 2 8, 6] q BCH-code. (minimum distance 6 b/c we have a consecutive set of size 5)

13 Are they BCH codes? Q: are the codes from THEOREM 1A BCH-codes? Q: are the codes from THEOREM 1A cyclic? A: no, except when q is even (see below). Therefore, the codes of THEOREM 1A for q odd are not BCH-codes.

14 Are the codes of THEOREM 2 new? Sloane s X-construction yields [q 3 + 1, 8, q(q 2 q 1)] q codes (when applied to a class of BCH-cdes). These are the parameters of the dual of the codes of THEOREM 2. The exact relationship between these classes of codes has yet to be determined.

15 The Automorphism Groups The codes of THEOREM 1 and 2 have large automorphism groups. One can describe the automorphism groups as representations of groups: THEOREM 1 THEOREM 2 MAut : PΓ 2 L(2, q 2 ) PGL(8, q) PΓ 3 L(2, q 3 ) PGL(9, q) ΓAut : PΓL(2, q 2 ) PΓL(8, q) PΓL(2, q 3 ) PΓL(8, q)

16 NOTATION ϕ a,b,c,d : x ax + c bx + d an element in PGL(2, q t ) ad bc 0 with φ e : x x q (of order e) PΓ e L(k, q e ) = PGL(k, q e ) φ e

17 Cyclic Collineations Consider the columns of the generator matrix as the homogeneous coordinates of a set of points in PG(k 1, q). A code automorphism is a stabilizer of this set. A constacyclic code automorphism permutes this set cyclically. A cyclic code automorphism permutes this set cyclically and maps the first vector to itself (i.e., no scalar multiplication allowed) after n applications.

18 Cyclic Collineations That is, we have to study the equations T n v = v, and T i v v i = 1,..., n 1 (for cyclic codes) and T n v = cv, and T i v v i = 1,..., n 1, c 0 (for constacyclic codes). Here T is the matrix associated to a collineation of PG(k 1, q). The generator matrix is obtained by writing T i v (i = 0, 1,..., n 1) into the columns of a matrix.

19 Cyclic Collineations Since our codes are obtained as images of the projective line, we are interested in cyclic (and constacyclic) automorphisms of PG(1, q t ). More generally, we wish to determine the collineations which permute the θ d 1 (q) := qd 1 q 1 points of PG(d 1, q) in one cycle (a.k.a. Singer cycle)

20 Cyclic Collineations LEMMA: (Hirschfeld 1973) # cyclic projectivities of PG(n, q) = # subprimitive polynomials of degree d over F q = (q 1) Φ(θ d q(q)) d (with Φ Euler s totient function)

21 Cyclic Collineations NOTE: If m(x) = x n+1 + c n x n + + c 0 F q [x] with c 0 0 and c = (c 1,..., c n ) then ( ) 0n c T m := 0 c I n induces a collineation whose characteristic poynomial is m(x).

22 Exponent and Subexponent Let m(x) F q [x] be monic, irreducible of degree d > 1. Exp(m) = smallest positive integer e such that m(x) divides x e 1 Subexp(m) = smallest poitive integer s such that m(x) divides x s c for some c F q (c is called integral element). m(x) is called primitive if exp(m) = q d 1 m(x) is called subprimitive if Subexp(m) = θ d 1 (q)

23 Exponent and Subexponent Subexp(m) = Exp(m) gcd(q 1, Exp(m)). b/c if β is a root of m(x) in F q d then s is the order of βf q in the factor group F /F q d q.

24 Cyclic Code Automorphisms For projective codes, the situation is as follows: { { cyclic collineation cyclic collineation with integral elt c = 1 } } { constacyclic code { } cyclic code }

25 Counting Irreducible Polynomials by Integral Element In order to count how many cyclic code automorphisms there are, we need to determine the number of irreducible polynomials with integral element 1. More generally, we will now determine the number of irreducible polynomials with integral element c F q.

26 Counting Irreducible Polynomials by Integral Element Notation: R c (d, q) = the set of monic irreducible polynomials of degree d over F q with integral element c F q R c (d, q) = #R c (d, q) θ d 1 (q) = r i=1 pe i i factorization of θ d 1 (q) into prime powers α a primitive element of F q.

27 Counting Irreducible Polynomials by Integral Element LEMMA: R α i (d, q) = 1 ( r ) ( r ) p e j d j Φ p e j j j=1 j=1 p j q 1 p j q 1 0 otherwise. if gcd(i, q 1, θ d 1 (q)) = 1, The function R α i (d, q) is periodic in i with period gcd(q 1, θ d 1 (q)). The non-zero function values depend only on d and q, but not on i.

28 Counting Irreducible Polynomials by Integral Element COROLLARY: 1 Φ(q + 1) for all c if q is even, R c (2, q) = 2 Φ(q + 1) if q is odd and c is a nonsquare in F q, 0 if q is odd and c is a nonzero square in F q. COROLLARY: R 1 (2, q) = { 1 2 Φ(q + 1) if q is even, 0 if q is odd.

29 Cyclic Code Automorphisms COROLLARY: The codes of length q or q are cyclic iff q is even COROLLARY: The codes of length q for odd q are not BCH-codes NOTE: If the codes are cyclic, then they are cyclic in R 1 (2, q) many ways.

30 The Representation Associated with Theorem 1 ϕ a,b,c,d U(a, b, c, d, β) = (U 1 U 2 U 3 ) with U i as follows (using β a primitive elt of F q 2 and δ = 1/(β β q ) and γ = βδ) 0 U 1 = N 2 (d 2 ) 4N 2 (bd) N 2 (b 2 1 ) N 2 (ad) N 2 (cd) +N 2 (bc) N 2 (ab) +T 2 (a q bcd q ) N 2 (c 2 ) 4N 2 (ac) N 2 (a 2 ) T 2 (cd 2q+1 2T ) 2 (ab q d q+1 ) +2T 2 (b q+1 cd q T ) 2 (ab 2q+1 ) T 2 (cd 2q+1 2T β) 2 (ab q d q+1 β) +2T 2 (b q+1 cd q T β) 2 (ab 2q+1 β) T 2 (c 2 d 2q ) 4T 2 (ab q cd q ) T 2 (a 2 b 2q ) T 2 (c 2 d 2q β) 4T 2 (ab q cd q β) T 2 (a 2 b 2q β) T 2 (c q+2 d q 2T ) 2 (a q+1 cd q ) +2T 2 (ab q c q+1 T ) 2 (a q+2 b q ) C T 2 (c q+2 d q 2T β) 2 (a q+1 cd q β) A +2T 2 (ab q c q+1 T β) 2 (a q+2 b q β)

31 The Representation Associated with Theorem 1 0 U 2 = 2T 2 (b q d q+2 γ) 2T 2 (bd 2q+1 δ) T 2 (b 2q d 2 γ) T 2 (a q cd q+1 γ) +T 2 (b q c q+1 dγ) T 2 (ac q d q+1 δ) +T 2 (bc q+1 d q δ) T 2 (a q b q cdγ) 2T 2 (a q c q+2 γ) 2T 2 (ac 2q+1 δ) T 2 (a 2q c 2 γ) 2T 2 (b q cd q+1 γ) 2T 2 (bc q d q+1 δ) +T 2 (a q d q+2 γ) +T 2 (ad 2q+1 T δ) 2 (a q b q d 2 γ) +T 2 (b q c q d 2 γ) +T 2 (bcd 2q +T δ) 2 (b 2q cdγ) 2T 2 (b q cd q+1 βγ) +T 2 (a q d q+2 β q γ) +T 2 (b q c q d 2 β q γ) 2T 2 (a q c q d 2 γ) +2T 2 (b q c 2 d q γ) 2T 2 (a q c q d 2 β q γ) +2T 2 (b q c 2 d q βγ) 2T 2 (a q c q+1 dγ) +T 2 (a q c 2 d q γ) +T 2 (b q c q+2 γ) 2T 2 (a q c q+1 dβ q γ) +T 2 (a q c 2 d q βγ) +T 2 (b q c q+2 βγ) 2T 2 (bc q d q+1 β q δ) +T 2 (ad 2q+1 βδ) +T 2 (bcd 2q βδ) 2T 2 (bc 2q dδ) +2T 2 (acd 2q δ) 2T 2 (bc 2q dβ q δ) +2T 2 (acd 2q βδ) 2T 2 (ac q+1 d q δ) +T 2 (ac 2q dδ) +T 2 (bc 2q+1 δ) 2T 2 (ac q+1 d q βδ) +T 2 (ac 2q dβ q δ) +T 2 (bc 2q+1 β q δ) T 2 (a q b q d 2 β q γ) +T 2 (b 2q cdβγ) T 2 (a 2q d 2 γ) +T 2 (b 2q c 2 γ) T 2 (a 2q d 2 β q γ) +T 2 (b 2q c 2 βγ) T 2 (a 2q cdγ) +T 2 (a q b q c 2 γ) T 2 (a 2q cdβ q γ) +T 2 (a q b q c 2 βγ) 1 C A

32 The Representation Associated with Theorem 1 0 U 3 = T 2 (b 2 d 2q δ) 2T 2 (b 2q+1 dγ) 2T 2 (b q+2 d q δ) T 2 (abc q d q δ) T 2 (a q+1 b q dγ) +T 2 (a q b q+1 cγ) T 2 (ab q+1 c q δ) +T 2 (a q+1 bd q δ) T 2 (a 2 c 2q δ) 2T 2 (a 2q+1 cγ) 2T 2 (a q+2 c q δ) T 2 (b 2 c q d q 2T δ) 2 (a q b q+1 dγ) 2T 2 (ab q+1 d q δ) +T 2 (abd 2q +T δ) 2 (ab 2q dγ) +T 2 (a q b 2 d q δ) +T 2 (b 2q+1 cγ) +T 2 (b q+2 c q δ) T 2 (b 2 c q d q β q δ) +T 2 (abd 2q βδ) T 2 (a 2 d 2q δ) +T 2 (b 2 c 2q δ) T 2 (a 2 d 2q βδ) +T 2 (b 2 c 2q β q δ) T 2 (abc 2q δ) +T 2 (a 2 c q d q δ) T 2 (abc 2q β q δ) +T 2 (a 2 c q d q βδ) 2T 2 (a q b q+1 dβ q γ) +T 2 (ab 2q dβγ) +T 2 (b 2q+1 cβγ) 2T 2 (a 2q bdγ) +2T 2 (ab 2q cγ) 2T 2 (a 2q bdβ q γ) +2T 2 (ab 2q cβγ) 2T 2 (a q+1 b q cγ) +T 2 (a 2q+1 dγ) +T 2 (a 2q bcγ) 2T 2 (a q+1 b q cβγ) +T 2 (a 2q+1 dβ q γ) +T 2 (a 2q bcβ q γ) 2T 2 (ab q+1 d q βδ) +T 2 (a q b 2 d q β q δ) +T 2 (b q+2 c q β q δ) 2T 2 (a q b 2 c q δ) +2T 2 (a 2 b q d q δ) 2T 2 (a q b 2 c q β q δ) +2T 2 (a 2 b q d q βδ) 2T 2 (a q+1 bc q δ) +T 2 (a q+2 d q δ) +T 2 (a 2 b q c q δ) 2T 2 (a q+1 bc q β q δ) +T 2 (a q+2 d q βδ) +T 2 (a 2 b q c q βδ) 1 C A

33 The Representation Associated with Theorem 2 ϕ a,b,c,d U(a, b, c, d, β) = (U 1 U 2 ) diag(1, 1, δ 1, δ 1, δ 1, δ 2, δ 2, δ 2 ) with U i as follows (using β a primitive elt of F q 3) U 1 = N 3 (d) N 3 (b) T 3 (m dbd γ 3 ) T 3 (m dbd γ 4 ) N 3 (c) N 3 (a) T 3 (m cac γ 3 ) T 3 (m cac γ 4 ) T 3 (m ddc ) T 3 (m bba ) T 3 (η 1,1 γ 3 ) T 3 (η 1,1 γ 4 ) T 3 (m ddc β) T 3 (m bba β) T 3 (η 1,2 γ 3 ) T 3 (η 1,2 γ 4 ) T 3 (m ddc β 2 ) T 3 (m bba β 2 ) T 3 (η 1,3 γ 3 ) T 3 (η 1,3 γ 4 ) T 3 (m dcc ) T 3 (m baa ) T 3 (ζ 1,1 γ 3 ) T 3 (ζ 1,1 γ 4 ) T 3 (m dcc β 11 ) T 3 (m baa β 11 ) T 3 (ζ 1,2 γ 3 ) T 3 (ζ 1,2 γ 4 ) T 3 (m dcc β 22 ) T 3 (m baa β 22 ) T 3 (ζ 1,3 γ 3 ) T 3 (ζ 1,3 γ 4 ),

34 The Representation Associated with Theorem 2 U 2 = T 3 (m dbd γ 5 ) T 3 (m bbd γ 6 ) T 3 (m bbd γ 7 ) T 3 (m bbd γ 8 ) T 3 (m cac γ 5 ) T 3 (m aac γ 6 ) T 3 (m aac γ 7 ) T 3 (m aac γ 8 ) T 3 (η 1,1 γ 5 ) T 3 (η 2,1 γ 6 ) T 3 (η 2,1 γ 7 ) T 3 (η 2,1 γ 8 ) T 3 (η 1,2 γ 5 ) T 3 (η 2,2 γ 6 ) T 3 (η 2,2 γ 7 ) T 3 (η 2,2 γ 8 ) T 3 (η 1,3 γ 5 ) T 3 (η 2,3 γ 6 ) T 3 (η 2,3 γ 7 ) T 3 (η 2,3 γ 8 ) T 3 (ζ 1,1 γ 5 ) T 3 (ζ 2,1 γ 6 ) T 3 (ζ 2,1 γ 7 ) T 3 (ζ 2,1 γ 8 ) T 3 (ζ 1,2 γ 5 ) T 3 (ζ 2,2 γ 6 ) T 3 (ζ 2,2 γ 7 ) T 3 (ζ 2,2 γ 8 ) T 3 (ζ 1,3 γ 5 ) T 3 (ζ 2,3 γ 6 ) T 3 (ζ 2,3 γ 7 ) T 3 (ζ 2,3 γ 8 ),

35 The Representation Associated with Theorem 2 with ( ) ( ) η1,1 η 1,2 η 1,3 mdbc m = dad m cbd 1 β β 2 1 β η 2,1 η 2,2 η 2,3 m bbc m bad m 10 β 20, abd 1 β 100 β 200 ( ) ( ) ζ1,1 ζ 1,2 ζ 1,3 mdac m = cad m cbc 1 β 11 β 22 1 β ζ 2,1 ζ 2,2 ζ 2,3 m bac m aad m 110 β 220, abc 1 β 101 β 202 γ 3 = β 102 β 201, γ 4 = β 200 β 2, γ 5 = β β 100, γ 6 = β 123 β 213, γ 7 = β 202 β 22, γ 8 = β 11 β 101, δ 1 = 1/ (T 3 (β 21 β 12 )), δ 2 = 1/ (T 3 (β 123 β 132 )), using the conventions that m xyz = x q2 y q z, β ijk = β iq2 +jq+k, β jk = β 0jk, and β k = β 00k.