Finite dihedral group algebras and coding theory

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1 Finite dihedral group algebras and coding theory César Polcino Milies Universidade de São Paulo

2 Basic FActs The basic elements to build a code are the following:

3 Basic FActs The basic elements to build a code are the following: A finite set, A called the alphabet. We shall denote by q = A the number of elements in A.

4 Basic FActs The basic elements to build a code are the following: A finite set, A called the alphabet. We shall denote by q = A the number of elements in A. Finite sequences of elements of the alphabet, that are called words. The number of elements in a word is called its length. We shall only consider codes in which all the words have the same length n.

5 Basic FActs The basic elements to build a code are the following: A finite set, A called the alphabet. We shall denote by q = A the number of elements in A. Finite sequences of elements of the alphabet, that are called words. The number of elements in a word is called its length. We shall only consider codes in which all the words have the same length n. A q-ary block code of length n is any subset of the set of all words of length n, i.e., the code C is a subset: C A n = A } A {{ A }. n veces

6 A classical scheme due to Shannon Information codification signal channel noise signal decodification receiver

7 A classical scheme due to Shannon Information codification signal channel noise signal decodification receiver The basic idea in coding theory, is to add information to the message, called redundancy, in such a way that it will turn possible to detect errors and correct them.

8 Definition Given two elements x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in A n, the number of coordinates in which the two elements differ is called the Hamming distance from x to y; i.e.: d(x, y) = {i x i y i, 1 i n}

9 Definition Given two elements x = (x 1, x 2,..., x n ) and y = (y 1, y 2,..., y n ) in A n, the number of coordinates in which the two elements differ is called the Hamming distance from x to y; i.e.: d(x, y) = {i x i y i, 1 i n} Definición Given a code C A n the minimum distance of C is the number: d = min{d(x, y) x, y C, x y }.

10 Theorem Let C be a code with minimum distance d and set [ ] d 1 κ = 2 where [x] denotes the integral part of the real number x; i.e., the greatest integer smaller than or equal to x. Then C is capable of detecting n 1 errors and correcting κ errores.

11 Theorem Let C be a code with minimum distance d and set [ ] d 1 κ = 2 where [x] denotes the integral part of the real number x; i.e., the greatest integer smaller than or equal to x. Then C is capable of detecting n 1 errors and correcting κ errores. Definition The number κ is called the capacity of the code C.

12 Linear Codes Basic Facts We shall take, as an alphabet A, a finite field F.

13 Linear Codes Basic Facts We shall take, as an alphabet A, a finite field F. In this case, F n is an n-dimensional vector space over F.

14 Linear Codes Basic Facts We shall take, as an alphabet A, a finite field F. In this case, F n is an n-dimensional vector space over F. We shall take, as codes, subespaces of F n of dimensión m < n.

15 Linear Codes Basic Facts We shall take, as an alphabet A, a finite field F. In this case, F n is an n-dimensional vector space over F. We shall take, as codes, subespaces of F n of dimensión m < n. Definition A code C as above is called a linear code over F. If d the minimum distance of C, we shall call it a (n,m,d)-code.

16 Definition A linear code C F n is called a cyclic code if for every vector (a 0, a 1,..., a n 2, a n 1 ) in the code, we have that also the vector (a n 1, a 0, a 1,..., a n 2 ) is in the code.

17 Definition A linear code C F n is called a cyclic code if for every vector (a 0, a 1,..., a n 2, a n 1 ) in the code, we have that also the vector (a n 1, a 0, a 1,..., a n 2 ) is in the code. Notice that the definition implies that if (a 0, a 1,..., a n 2, a n 1 ) is in the code, then all the vectors obtained from this one by a cyclic permutation of its coordinates are also in the code.

18 Let R n = F[X ] X n 1 ; We shall denote by [f ] the class of the polynomial f F[X ] in R n.

19 Let R n = F[X ] X n 1 ; We shall denote by [f ] the class of the polynomial f F[X ] in R n. The mapping: ϕ : F n F[X ] X n 1 (a 0, a 1,..., a n 2, a n 1 ) F[X ] [a 0 + a 1 X a n 2 X n 2 + a n 1 X n 1 ].

20 Let R n = F[X ] X n 1 ; We shall denote by [f ] the class of the polynomial f F[X ] in R n. The mapping: ϕ : F n F[X ] X n 1 (a 0, a 1,..., a n 2, a n 1 ) F[X ] [a 0 + a 1 X a n 2 X n 2 + a n 1 X n 1 ]. ϕ is an isomorphism of F-vector spaces. Hence A code C F n is cyclic if and only if ϕ(c) is an ideal of R n.

21 In the case when C n = a a n = 1 = {1, a, a 2,..., a n 1 } is a cyclic group of order n, and F is a field, the elements of FC n are of the form: α = α 0 + α 1 a + α 2 a α n 1 a n 1.

22 In the case when C n = a a n = 1 = {1, a, a 2,..., a n 1 } is a cyclic group of order n, and F is a field, the elements of FC n are of the form: It is easy to show that α = α 0 + α 1 a + α 2 a α n 1 a n 1. FC n = Rn = F[X ] X n 1 ;

23 In the case when C n = a a n = 1 = {1, a, a 2,..., a n 1 } is a cyclic group of order n, and F is a field, the elements of FC n are of the form: It is easy to show that α = α 0 + α 1 a + α 2 a α n 1 a n 1. FC n = Rn = F[X ] X n 1 ; Hence, to study cyclic codes is equivalent to study ideals of a group algebra of the form FC n.

24 Definition A group code is an ideal of a finite group algebra.

25 Definition A group code is an ideal of a finite group algebra. If we assume that char(f) G, then to study group codes is equivalent to study ideals in group algebras, generated by idempotent elements

26 Idempotents from subgroups Let H be a subgroup of a finite group G and let F be a field such that car(f) G. The element e H = 1 H h h H is an idempotent of the group algebra FG, called the idempotent determined by H.

27 Idempotents from subgroups Let H be a subgroup of a finite group G and let F be a field such that car(f) G. The element e H = 1 H h h H is an idempotent of the group algebra FG, called the idempotent determined by H. e H is central if and only if H is normal in G.

28 If H is a normal subgroup of a group G, we have that FG Ĥ = F[G/H].

29 If H is a normal subgroup of a group G, we have that FG Ĥ = F[G/H]. so ) dim F ((FG) Ĥ = G H = [G : H].

30 If H is a normal subgroup of a group G, we have that FG Ĥ = F[G/H]. so ) dim F ((FG) Ĥ = G H = [G : H]. Set τ = {t 1, t 2,..., t k } a transversal of K in G (where k = [G : H] and we choose t 1 = 1),

31 If H is a normal subgroup of a group G, we have that FG Ĥ = F[G/H]. so ) dim F ((FG) Ĥ = G H = [G : H]. Set τ = {t 1, t 2,..., t k } a transversal of K in G (where k = [G : H] and we choose t 1 = 1), then is a a basis of (FG) Ĥ. {t i Ĥ 1 i k}

32 Is it possible to determine the primitive central idempotents from the idempotents determined by subgroups?

33 Theorem (Arora-Pruthi (1997), Ferraz-P.M. (2007)) Let F be a field with q elements and A a cyclic group of order p n such that o(q) = ϕ(p n ) in U(Z p n) (where ϕ denots Euler s Totient function). Let A = A 0 A 1 A n = {1} be the descending chain of all subgroups of A. Then, the set of primitive idempotents of FA is given by: ) e 0 = 1 p n ( a A a e i = Âi Âi 1, 1 i n.

34 Theorem (Arora and Pruthi (2002), Ferraz-PM (2007)) Let F be a field with q elements and A a cyclic group of order 2p n, p an odd prime, such that o(q) = ϕ(p n ) in U(Z 2p n). Write G = C A where A denotes the p-sylow subgroup of G and C = {1, t} is the 2-Sylow subgroup. If e i, 0 i n denotes the set of primitive idempotents of FA, then the primitive idempotents of FG are (1 + t) 2 e i and (1 t) 2 e i 0 i n.

35 Let A be an abelian p-group. For each subgroup H of A such that A/H {1} is cyclic, we shall construct an idempotent of FA. Since A/H is a cyclic subgroup of order a power of p, there exists a unique subgroup H of A, containing H, such that H /H = p.

36 Let A be an abelian p-group. For each subgroup H of A such that A/H {1} is cyclic, we shall construct an idempotent of FA. Since A/H is a cyclic subgroup of order a power of p, there exists a unique subgroup H of A, containing H, such that H /H = p. We set e H = Ĥ Ĥ. and also e G = 1 g. G g G

37 Theorem (Ferraz-PM (2007)) Let p be an odd prime and let A be an Abelian p-group of exponent p r. Then, the set of idemponts above is the set of primitive idempotents of FA if and only if one of the following holds: (i) p r = 2, and q is odd. (ii) p r = 4 and q 3 (mod 4). (iii) o(q) = ϕ(p n ) in U(Z p n).

38 Theorem (Ferraz-PM (2007)) Let p be an odd prime and let A be an Abelian p-group of exponent p r. Then, the set of idemponts above is the set of primitive idempotents of FA if and only if one of the following holds: (i) p r = 2, and q is odd. (ii) p r = 4 and q 3 (mod 4). (iii) o(q) = ϕ(p n ) in U(Z p n). Theorem (Ferraz-PM (2007)) Let p be an odd prime and let A be an abelian p-group of exponent 2p r. Write A = E B, where E is an elementary abelian 2-group and B a p-group. Then the primitive idempotents of FA are products of the form e.f, where e is a primitive idempotent of FE and f a primitive idempotent of FB.

39 Theorem (Ferraz-PM (2007)) Let F be a finite field with F = q, and let A be a finite abelian group, of exponent e. Then the primitive central idempotents can be constructed as above if and only if one of the following holds: (i) e = 2 and q is odd. (ii) e = 4 and q 3 (mod 4). (iii) e = p n and o(q) = ϕ(p n ) in U(Z p n). (iv) e = 2p n and o(q) = ϕ(p n ) in U(Z 2p n).

40 Let F be a finite field with q elements and let D n be the dihedral group of order 2n, i.e. D n = a, b a n = b 2 = 1, bab = a 1.

41 Let F be a finite field with q elements and let D n be the dihedral group of order 2n, i.e. D n = a, b a n = b 2 = 1, bab = a 1. We shall assume throughout that char(f) does not divide 2n.

42 Lemma Let F be a finite field and let ζ be an e th -root of unity, where e is the exponent of D n. Then, the number of simple components of FD n equals the number of simple components of QD n if and only if denoting by q the residue class of q in Z n, we have that either q = U(Z n ) or q is a subgroup of index 2 in U(Z n ) and 1 q.

43 Theorem (Ferraz, Dutra, PM) The idempotents of FD n can be written from idempotents determined by subgroups (in a way similar to that of the abelian case) if and only if one of the following holds: (i) n = 2 or 4 and q is odd. (ii) n = 2 m, with m 3 and q is congruent to either 3 or 5, modulo 8. (iii) n = p m with p an odd prime and the class q is a generator of the group U(Z p m). (iv) n = p m with p an odd prime, the class q is a generator of the group U 2 (Z p m) = {x 2 x U(Z p m}) and 1 is not a square modulo p m.

44 (v) n = 2p m with p an odd prime and the class q is a generator of the group U(Z 2p m). (vi) n = 2p m with p an odd prime, the class q is a generator of the group U 2 (Z p m) = {x 2 x U(Z 2p m}) and 1 is not a square modulo 2p m. (vii) n = 4p m with p an odd prime and both q and q have order ϕ(p m ) modulo 4p m. (viii) n = p m 1 1 pm 2 2 with p 1, p 2 odd primes, (ϕ(p m 1 1 ), ϕ(pm 2 2 )) = 2 and both q and q have order ϕ(p m 1 1 )ϕ(pm 2 2 )/2 modulo pm 1 1 pm 2 2. (ix) n = 2p m 1 1 pm 2 2 with p 1, p 2 odd primes, (ϕ(p m 1 1 ), ϕ(pm 2 2 )) = 2 and both q and q have order ϕ(p m 1 1 )ϕ(pm 2 2 )/2 modulo p m 1 1 pm 2 2.

45 Let Q n with n even be a generalized quaternion group; i.e, a group given by the presentation: with m 2. Q n = α, β α n = 1 β 2 = α n/2, β 1 αβ = α 1

46 Let Q n with n even be a generalized quaternion group; i.e, a group given by the presentation: with m 2. Q n = α, β α n = 1 β 2 = α n/2, β 1 αβ = α 1 Theorem The group algebras FD n and FQ n are isomorphic if and only if 4 n or q 1 (mod 4).

47 Code Parameters Theorem Let G be a finite group and let F be a field such that char(f) G. Let H and H be normal subgroups of G such that H H and set e = Ĥ Ĥ. Then, dim F (FG)e = G/H G/H = G ( 1 H ) H H and w((fg)e) = 2 H where w((fg)e) denotes the minimal distance of (FG)e.

48 Theorem Let G be a finite group and let F be a field such that char(f) G. Let H and H be normal subgroups of G such that H H and set e = Ĥ Ĥ. Let A be a transversal of H in G and τ a transversal of H in H containing 1. Then is a basis of (FG)e over F. B = {a(1 t)ĥ a A, t τ \ {1}}

49 Theorem Let H i Hi, be normal subgroups of a group G, 1 i k, such that Hi Ni = {1}, where N i denotes the subgroup generated by all Hj with j i. Set e = (Ĥ1 Ĥ 1 )(Ĥ2 Ĥ 2 ) (Ĥk Ĥ k ). Then, ( G dim F (FG)e = 1 H ) ( 1 H 1 H 2 H k H1 1 H ) ( 2 H2 1 H ) k Hk and w ((FG)e) = 2 k H 1 H 2 H k.

50 Theorem Let H i Hi, be normal subgroups of a group G, 1 i k, such that Hi Ni = {1}, where N i denotes the subgroup generated by all Hj with j i. Set e = (Ĥ1 Ĥ 1 )(Ĥ2 Ĥ 2 ) (Ĥk Ĥ k ). Let A be a transversal of H in G and τ i a transversal of H i in Hi containing 1, 1 i k. Then B = {a(1 t 1 )(1 t 2 ) (1 t k )Ĥ a A, t i τ i, t i 1, 1 i k} is a basis of (FG)e over F.

51 Dimensions and weights when n involves only one prime. n e dim[(fd n )e] w[(fd n )e] 4 1 a m e 2 i = Ĉ2 i Ĉ 2 i+1 2 m i 2 i+1 p m e p i = Ĉp i Ĉ p i+1 2ϕ(p i i ) 2p i

52 Dimensions and weights when n involves two primes. n dim[i] w[i] 2p m e 2 = Ĉ2 e p i = Ĉp i Ĉ p i+1 2ϕ(p m i ) 4p i e 2 = 1 Ĉ2 e p = Ĉpm 2 2pi e 2 = 1 Ĉ2 e p i = Ĉp i Ĉ p i+1 2ϕ(p m i ) 4p i 4p m e = Ĉ4 e i p = Ĉp i Ĉ p i+1 2ϕ(p m i ) 8p i e 2 i = Ĉ2 i Ĉ 2 i+1, e p = Ĉp m ϕ(2i ) 2. 2 i p m e 2 i = Ĉ2 i Ĉ 2 i+1, e p i = Ĉp i Ĉ p i+1 2ϕ(2 i )ϕ(p j ) 4. 2 i p j p m 1 1 p m 2 2 e p1 = Ĉ m p 1 1 e p i = Ĉp i Ĉ 1 1 p i+1 1 e p i = Ĉp i Ĉ 1 1 p i+1 1 e j = Ĉp j Ĉ j+1 p p e p2 = Ĉ m p 2 2 e j = Ĉp j Ĉ j+1 p p ϕ(p m 2 j 2 ) 2p m 1 1 p j 2 2ϕ(p m 1 i 1 ) 2p1 i pm 2 2 2ϕ(p m 1 i 1 )ϕ(p m 2 j 2 ) 4p1 i pj 2

53 Dimensions and weights when n = 2p m 1 1 pm 2 2. e e 1 e 2 dim[fd n)ee 1 e 2 ] w[(fd n)ee 1 e 2 ] Ĉ 2 Ĉ m 1 p 1 Ĉ 2 Ĉ p i Ĉ 1 p i Ĉ2 Ĉ m p Ĉ2 Ĉ p i Ĉ j+i 1 p 1 Ĉ 2 Ĉ p i Ĉ j+i 1 p 1 1 Ĉ2 Ĉ p i Ĉ j+i 1 p 1 Ĉ j Ĉ j+1 p p 2 2 C m 2 p 2 Ĉ j Ĉ j+1 p p 2 2 Ĉ m p 2 2 Ĉ j Ĉ j+1 p p 2 2 Ĉ p j 2 Ĉ p j+1 2 2ϕ(p m 2 j 2 ) 4p m 1 1 p j 2 2ϕ(p m 1 i 1 ) 4p1 i pm 2 2 2ϕ(p m 2 j 2 ) 4p m 1 1 p j 2 2ϕ(p m 1 i 1 ) 4p1 i pm 2 2 2ϕ(p m 1 i 1 ϕ(p m 2 j 2 ) 8p i 1 pj 2 2ϕ(p m 1 i 1 ϕ(p m 2 j 2 ) 8p i 1 pj 2

54 THANK YOU!!

Essential idempotents in group algebras and in Coding Theory

Essential idempotents in group algebras and in Coding Theory in group algebras and in Coding Theory César Polcino Milies Universidade de São Paulo and Universidade Federal do ABC NON COMMUTATIVE RINGS AND APPLICATIONS IV, LENS, 2015 Definition A linear code C F