CC BY
17
ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 3.
УДК 512.552.7+519.725 DOI 10.22405/2226-8383-2019-20-3-107-123
Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения
К. В. Веденёв, В. М. Деундяк
Веденёв Кирилл Владимирович — Южный федеральный университет (г. Ростов-на-Дону).
e-mail: [email protected]
Деундяк Владимир Михайлович — кандидат физико-математических наук, доцент, Южный федеральный университет, ФГАНУ НИИ «Спецвузавтоматика» (г. Ростов-на-Дону). e-mail: [email protected],
Аннотация
В 1978 году Р. Мак-Элисом построена первая асимметричная кодовая криптосистема, основанная на применении помехоустойчивых кодов Гоппы, при этом эффективные атаки на секретный ключ этой криптосистемы до сих пор не найдены. К настоящему времени известно много криптосистем, основанных на теории помехоустойчивого кодирования. Одним из способов построения таких криптосистем является модификация криптосистемы Мак-Элиса с помощью замены кодов Гоппы на другие классы кодов. Однако, известно что криптографическая стойкость многих таких модификаций уступает стойкости классической криптосистемы Мак-Элиса.
В связи с развитием квантовых вычислений кодовые криптосистемы, наряду с крипто-системамми на решётках, рассматриваются как альтернатива теоретико-числовым. Поэтому актуальна задача поиска перспективных классов кодов, применимых в криптографии. Представляется, что для этого можно использовать некоммутативные групповые коды, т. е. левые идеалы в конечных некоммутативных групповых алгебрах.
Для исследования некоммутативных групповых кодов полезной является теорема Вед-дерберна, доказывающая существование изоморфизма групповой алгебры на прямую сумму матричных алгебр. Однако конкретный вид слагаемых и конструкция изоморфизма этой теоремой не определены, и поэтому для каждой группы стоит задача конструктивного описания разложения Веддерберна. Это разложение позволяет легко получить все левые идеалы групповой алгебры, т.е. групповые коды.
В работе рассматривается полупрямое произведение Qm,n = (Zm х Z„) X (Z2 х Z2) абелевых групп и конечная групповая алгебра Fg Qm,n этой группы. Для этой алгебры при условиях п | q — 1 и НОД(2топ, q) = 1 построено разложение Веддербёрна. В случае поля чётной характеристики, когда эта групповая алгебра не является полупростой, также получена сходная структурная теорема. Описаны все неразложимые центральные идемпотенты этой групповой алгебры. Полученные результаты используются для алгебраического описания всех групповых кодов над Qm,„.
Ключевые слова: групповая алгебра, полупрямое произведение, конечное поле, разложение Веддербёрна, левые идеалы, групповые коды.
Библиография: 21 названий. Для цитирования:
К. В. Веденёв, В. М. Деундяк. Структура конечной групповой алгебры одного полупрямого произведения абелевых групп и её приложения // Чебышевский сборник. 2019. Т. 20, вып. 3, с. 107-123.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 3.
UDC 512.552.7+519.725 DOI 10.22405/2226-8383-2019-20-3-107-123
The structure of finite group algebra of a semidirect product of abelian groups and its applications
K. V. Vedenev, V. M. Deundvak
Vedenev Kirill Vladimirovich — Southern Federal University (Rostov-on-Don). e-mail: [email protected]
Deundyak Vladimir Mikhailovich — candidate of physical and mathematical Sciences, associate Professor, Southern Federal University, Research Institute "Specvuzavtomatika" (Rostov-on-Don). e-mail: [email protected],
Abstract
In 1978 R. McEliece developed the first assymetric cryptosystem based on the use of Goppa's error-correctring codes and no effective key attacks has been described yet. Now there are many code-based cryptosystems known. One way to build them is to modify the McEliece cryptosystem by replacing Goppa's codes with other codes. But many variants of this modification were proven to be less secure.
In connection with the development of quantum computing code cryptosystems along with lattice-based cryptosystems are considered as an alternative to number-theoretical ones. Therefore, it is relevant to find promising classes of codes that are applicable in cryptography. It seems that for this non-commutative group codes, i.e. left ideals in finite non-commutative group algebras, could be used.
The Wedderburn theorem is useful to study non-commutative group codes. It implies the existence of an isomorphism of a semisimple group algebra onto a direct sum of matrix algebras. However, the specific form of the summands and the isomorphism construction are not explicitly-defined by this theorem. Hence for each semisimple group algebra there is a task to explicitly construct its Wedderburn decomposition. This decomposition allows us to easily describe all left ideals of group algebra, i.e. group codes.
In this paper we consider one semidirect product Qm,n = (Zm x Z„) X (Z2 x Z2) of abelian groups and the group algebra FgQm,n. In the case when n | q — 1 Mid gcd(2mn,q) = 1, the Wedderburn decomposition of this algebra is constructed. In the case when field is of 2
is also obtained. Further in the paper, the primitive central idempotents of this group algebra are described. The obtained results are used to algebraically describe the group codes over Qm,n.
Keywords: group algebra, semidirect product, finite field, Wedderburn decomposition, left ideals, group codes.
Bibliography: 21 titles. For citation:
K. V. Vedenev, V. M. Deundyak, 2019, "The structure of finite group algebra of a semidirect product of abelian groups and its applications" , Chebyshevskii sbornik, vol. 20, no. 3, pp. 107-123.
Introduction
Let G be a finite group with the identity e, written multiplicatively, let R be a ring with the identity 1r and Fg be a Galois field of order g. Recall, the group ring RG is a set of all formal linear combinations a = ^geG agg, ag £ R, equipped with operations of addition and (left and right) multiplication by elements of R defined componentwise and multiplication defined as follows:
^ 9 ^ Pgg = ^ ^ ®gh-1 Ph\ 9.
g£G g£G g£G \h£H )
(see [1]). In the case when R is commutative, RG is also called group algebra of G over R ([2], [1]). Note that, the correspondences g ^ iRg, g £ ^^d r ^ re, r £ R, define natural embeddings of the group G and the ring R into RG.
Any left ideal I C FgG is called a group code over G (see [3], [4]). This algebraic approach to codding theory was introduced by S.D. Berman [5]. In this approach, all elements of the field Fg are the encoding alphabet and the order of the group G is the length of codewords. Note that the dimension of a code C C Fg is its dimension as an Fg-subspace in FgG. Many classical codes can be realized as (left) ideals in group algebras (see survey [3]), including Reed-Solomon codes ([4], [6]) and Reed-Muller codes ([4], [5], [7]). Algebraic approach to error-correcting codes gives some benefits, i.e. additional algebraic structure helps to study more efficient encoding and decoding algorithms for known codes (see for example [8]) and to discover new classes of codes in group algebras ([9], [10], [11]).
Another motivation to study codes in non-commutative group algebras is that this codes could be useful in cryptography. R. McEliece developed an asymmetric cryptosvstem based on the use of binary Goppa codes in 1978 and no effective key attacks has been described yet. Code cryptosvstems are considered as a potential replacement to number-theoretical ones in the connection with the development of quantum computing (see NIST-PQC competition [12]). The main disadvantage of the original McEliece cryptosvstem is that the private and public keys are very large matrices. To reduce the key size there have been attempts to replace Goppa codes with other classes of error-correcting codes. Variants of the McEliece cryptosvstem based on the use of well-known Reed-Solomon codes and Reed-Muller codes, which can be realized as two-sided ideals in some abelian group algebras, were proven to be less secure ([13], [14], [15]). So, non-commutative codes, which are one-sided (left) ideals in non-commutative group algebras, could be a good option to build new resistant and convenient in use cryptosvstems.
The Wedderburn theorem implies that if FgG is semisimple then FgG is isomorphic to a direct sum of matrix algebras over some extensions of the field Fg. This theorem is a very powerful tool to study the structure of non-commutative codes, but it gives no information about the summands and the isomorphism. So, for an arbitarv group algebra FgG there is a problem of constructing its Wedderburn decomposition. There are several results on how to construct the Wedderburn decomposition and central primitive idempotents known (see [16], [17] ). In [18] the Wedderburn decomposition of finite dihedral group algebra was described and in [11] this decomposition was used to study the dihedral codes.
Let m,n £ N and let Qmn be a group with the following presentation:
{a\,a2, b, c | a™, a%, b2, c2, a\ = a,ab2 = a-1, a1a2 = a2a1,bc = cb, ba1 = a1b, ca2 = a2c), (1)
hereinafter gg = g~xgg. We will call Qm,nthe (m, n)-bidihedral group. In this paper we consider the bidihedral group and its group algebra FgQm,n. Under certain conditions, we obtain its Wedderburn decomposition in the semisimple case. Also we prove the similar structure theorem in the non-semisimple case. Then we explicitly describe the primitive central idempotents of this algebra. Finally, the obtained results are applied to algebraic codding theory.
The paper is organized as follows. In section 1 we introduce some preliminaries about the bidihedral group Qm,n, its group algebra and polynomials over finite fields. In section 2 we prove the general structure theorem for this group algebra and then we obtain the Wedderburn decomposition of FqQm,n- In section 3 we construct the inverse of isomorphisms described in the previous section and then explicitly describe primitive central idempotents. In section 4 we apply this results to codding theory, i.e. we obtain the explicit description of the group codes over Qm,n.
1. Preliminaries
Let G be a group and S С G. Bellow, bv (S) we denote the subgroup of G generated bv S. Let D2n be a dihedral of order 2n, i.e. D2n has the presentation (see [19], p. 6):
D2n = (x,y | xn,y2,xy = x-1).
Consider the group Qm,n defined in (1). Hereinafter a1 ,a2, b, с are from (1).
Lemma 1. Let G1 = (a1,c) and G2 = (a2,b). Then
(i) G1 ~ D2m and G2 ~ D2n
(ii) Qm,n decomposes into a direct product of G1 and G2. Доказательство. We obviously have
G1 = (a-]_,c | a™, c2, a\ = a-1), G2 = (a2, b | b2, ab2 = a-1)
are presentations of G1 and G2. It follows that G1 ~ D2m and G2 ~ D2n.
Since [19], p. 3, it follows that a direct product of G1 and G2 has a presentation of the form (1), hence Qm,n decomposes into a direct product of G1 and G2. □ From previous lemma we obtain the following result.
Lemma 2. Let N = (a1,a2) and H = (b, c); then
(i) N ~ Zm x Zn is normal;
(ii) H ~ Z2 x Z2;
(Hi) Qm,n is a semidirect product of N by H (Qm,n = N X H).
Let R ^e a ring (field); bv Mn(R) we denote the ring (algebra) of (n x n)-matrices over R. Lemma 3. The group Qm,n is isomorphic to the matrix group
T —
'e1 nz1 mz2 0 £2 0 0 0 1
e Ms(Zmra) I €i = ±1, Z1, Z2 e Zn
Доказательство. Let
/1 n 0^ (H = I 0 1 0
\0 0 1
Observe that
'1 0 m> 0,2 = I 0 1 0 0 0 1
'-1 0 0^ b = I 0 -10 0 0 1;
=
'(-!)* (-1)tni (-1Утр
0 0
(-1)k+t 0
100 с= I 0 -1 0 001
It follows that a1, b2, b, a are the generators of Tm,n. It is easy to check that
af1 = a% = b2 = c2 = b°1, bf = a-1, = b-1,
b1b2 = b2b1, be = ab, ba1 = b1 b, ab2 = b2b. Hence we can define epimorphism (see [20], p. 15) <p : Qm,n ^ Tm,n by the generaters of Qm,n'
<p : a1^ a1, a2 ^ b2, b^b, c ^ b.
Since || = 4mn and |Tmn| = 4mn, it follows that <p is an isomorphism. □ Consider the group algebra FqQm,n .An v u £ FgQm,n can be written as
u = Po(a1,a,2) + bP1(a1,a,2) + cP2(a1,a,2) + bcP3(d1,112), (2)
where Pk(x1,x2) £ Fg[x1,x2\ has degree in x1 less than m and degree in x2 less than n, i.e. deg^ (Pfc) < m deg^2 (Pfc) < n.
Throughout this paper we will assume that gcd( mn, q) = 1 and n | q — 1.
Bellow we will use the following results on polynomials over finite fields. For every polynomial g(x) £ Fg[x] with g(0) = 0 9*(x) denotes its reciprocal polynomial, i.e., g*(x) = xdeg(a"lg(x-1). We say that a polinomial g(x) is auto-reciprocal if g(x) and g*(x) differ by a multiplicative constant. Define
1, n ,
£(n) := i 0
12, n is even.
The polynomials xm — 1 £ Fg [x^d xn — 1 £ Fg [x] split into monic irreducible factors as
xm — 1 = (h ... fri)(fr1 +1 fri + 11ri+2 fri+2 . . . fi"i+si fr1+s1), (3)
xn — 1 = (91... 9r2)(9r2+19*2+19r2+29*2+2 ... 9r2+s29**2+s2), (4)
where f1 = g1 = x — 1, f* = fj for 1 < j < r-^, g* = gj for 1 < j < r2\ and f2 = x + 1 if m is even, g2 = x + 1 if n is even. He re r1, r2 denote the numbers of auto-reciprocal factors in these factorizations and 2 s 1,2s2 denote the numbers of non-auto-reciprocal factors.
Since F* ~ Zq-1 and n | q — 1, it follows that there exist a multiplicative subgroup of F* of order n, hence the factors in (4) are of degree 1. And since x — 1 and x + 1 are the only auto-reciprocal polynomials of degree 1, it follows that r2 = £(n) and s2 = ,
Let h £ Fg[x] be irreducible, deg(h) = k and let a be a root of h in an extension of Fg. By Fg[a] we denote the extension of Fg with a. ft is well known that Fg [a] = Fg [a-1^d Fg [a] ~ Fqdeg(h). Any element t £ Fg[ a] can be written as v(a) or w(a-1), v,w £ Fg[x^d deg(v) < k, deg(u) < k. Polynomials v(x) and w(x) are called polynomial representations of t with a and a-1.
By ai we denote a root of the polynomial fj in an extension of Fg and by ftj we denote a root j
2. The Wedderburn decomposition of F qQmn
Bellow, by {h)k we denote the cyclic group of order k with a generator h. Let G be a group and R be a Fg-algebra, then we can extend multiplication by the elements of Fq to RG. Note that, RG equipped with this operation is a Fg-algebra.
For each i £ {1,..., r1 + Si^d j £ {1,..., r 2 + s 2} let ui,j be the Fg-algebras homomorphism of FqQm,n defined by the generators of Qm,n as follows:
1. 1 < i < ^(m^d 1 < j < r2:
^ Fg({hi}2 x {h2}2) Vi,j (ai) = on, (0,2) = Pj, Vi,j (b) = hi, Uij (c) = h2.
2. £(m) + 1 < i < ri + s^d 1 < j < r2:
^ M2(FgH)(h}2
vi,j (ai) = ( ^ a-i) , (a2) = (j! , (b) = ^ ^ h Vij (c) =
3. 1 < i < ^(m^d r2 + 1 < j < r2 + s2:
Vi,j • FgQm,n
^ M2(Fg){h}2
(11)
"U(ai) = (0 , "u(a2) = (0' p~l) , "u(6) = (? J) , "u(c) = (i 0)
h.
4. £(m) + 1 < i < ri + si and r2 + 1 < j < r2 + s2:
^ M4(Fq[a,])
Vi,j (ai) =
(on 0 0
0 ai 0
0 0
0 0 a-i 0
V0 0 0 a-iJ
(a2) =
0 p.
0 0
0
-i
j
0
0 0 Pi
0 0 0
0 0 p-iJ
v%,i (b) =
0100 1000 0001 0010
vi,i (c) =
0010 0 0 0 1 1000 0100
For each £(m) + 1 < i < ri define
* •= C ,
and automorphisms
Zi •=
1 0 -ai 0
0 1 0 - ai
1 0 -a-i 0
0 1 0 -a-y
a, • M2(Fg[a,])(h}2 ^ M2(Fg[a,])(h}2, al(X) = Z-iXZt;
al • M4(Fg[a,]) ^ M4(Fg[a,]), al(X) = Z~-ixZl.
Lemma 4. (i) Let £(m) + 1 < i < ri and 1 < j < r2; then im^z/jj) C M2(Fg[ai + a-i])(h}2. (ii) Let, £(m) + 1 < i < ri and r2 + 1 < j < r2 + s2; then im(<7jZ/jj) C M4(Fg[ai + a-i])-
Доказательство. In the case £(m) + 1 <i< ri + si and 1 < j < r2 we have
)(oi)= (-1 at + q"0 , (a'Vi-')(o2)=(o "J ' )(i>)=(0
)(«)=(0 -(Q,_+1Qr^.
Hence im(ajZ/j,j) C M2(Fg[ ai + ai i])(h}2-
Similar computations shows that in the case £(m) + 1 < i < ri + si and r2 + 1 < j < r2 + s 2 we have im( <7iZ/ij) C M4(Fg[ ai + a~i]). □
Замечание 1. Observe that, ift is a root of the polynomial g £ Fg[x], then t 1 is a root of g*. When g £ Fg[x] is auto-reciprocal and irreducible and g(1) = 0 5,(_1) = 0, there exists a polynomial h £ Fg[x]; such, that h(t + t_1) = 0 and deg(h) = de|(fl) (see [18], remark 3.2). It follows that
dimFq (Fq [t + t "i]) = degm. (5)
Finally, let us define the map
n+81 T2 , (C( m) + 1 < i < n) A (1 <j< r2)
P • = © © Pij, Pij •= \ , (£(m) + 1 < 1 < ri) A (r2 + 1 < j < r2 + S2) . (6)
%=i i=i , otherwise
Teopema 1. Let gcd(mn, q) = 1 and n \ q — 1; then the map
ri+Sl T2+S2
P • FqQm,n ^ CD (J) ,
i=i j=i
(
Bij =
Fg((hi)2 x (h2)2), (1 < i < £(m)) Л (1 < j < т2)
M2(Fq[Qi + Q~ 1])(h)2, (£(m) + 1 < i < n) Л (1 < j < Г2)
M2(Fg[Qi])(h)2, ( гi + 1 < г < n + si) Л (1 < j < Г2)
M2(Fg)(h)2, (1 < i < £(m)) Л (Г2 + 1 < j < Г2 + s 2)
M4(Fg[Qi + Q-i]), (£(m) + 1 < г < n) Л (Г2 + 1 <j< Г2 + S2)
M4(F,[Qi]), ( ri + 1 < i < n + si) Л ( Г2 + 1 < j < Г2 + S 2)
is an isomorphism.
Доказательство. First we show that p is injective, i.e. if p(u) = 0 then и = 0. Now let и £ FqQm,n be of the form (2) then
(1 < < ( m)) (1 < < 2)
(u) = Po(Qi) + Pi(Qi, Pj)hi + P2(Qi, Pj)h2 + Pa(Qi, Pj)hih2; (7)
2) in the case £(m) + 1 < г < ri + si and 1 < j < r2 we have
* (u) = (Po(Qi,Pj) P2(Q," 1,PjЛ + fPi(Qi,Pj) P3(Q,"iA h. (os
(U)=^P2(Q„P,) Po(Q" i,PJ ^4p3(Q»,P. ) PiK"i ,Pi V h; ^
1 < < ( m) 2 + 1 < < 2 + 2
j-1\\ /o c„, a \ r> c„, a-1
>P-1)) + \Ps(ai,Pj) P2(ai,P~1)/
) = /Po(ai, Pj) P1(aj,^-1)\ +(P2(ai,f3j) Ps(ai,^j-1A h m
(U) ) Pa^^V + ^ai^) P2(ai,Pj1)) h; W
( m) + 1 < < 1 + 1 2 + 1 < < 2 + 2
/ Pa(ai, Pj) P1(ai,P"1) P2(a-1,Pj) P3(a"1,P-1)\
^i,j(u) =
P1(ai,Pj) Pa(ai,Pj-1) P3(a-1,Pj) P2(a-1,PJ1) P2(ai,Pj) P3(ai, Pj-1) Pa(a-1,Pj) P^a-1^-1) \P3(ai, Pj) P2(ai, Pj-1) P1(a-1,Pj) Pa(al-1, Pj-1))
(10)
Note that, since ai £ Fg[ai] and Pj £ Fg, it follows that Pk(ai, P±l), Pk(a- 1 ,Pj±1) £ Fg[ai]. Since p(u) = 0 we see that Vij(u) = 0. It follows that
Pk(ai,P3) = Pk(ai, P-1) = Pi(a-\Pj) = Pl(a-1,P~1) = 0 (k = 0..3) (11)
for all 1 < i < r1 + s^ 1 < j < r2 + s2. Since dega,i Pk(x1,x2) < m and dega,2 Pk(x1,x2) < n, it follows that
m-1 n-1 m-1 fn-1 \ m-1
Pk(x1,x2) = ci,jx\xi = Y,x\ I ci,jx32 I = x\Pki(x2), degPki(x) < n.
i=a j=a i=a \j=a J i=a
Using (11) and (3), we obtain Pk(x,Pj) £ Fg[x] and Pk(x,P-1) £ Fg[x] are divisible by the polynomial xm — 1 to all j £ {1,...,r2 + s2}. Since degxi Pk(x1,x2) < m, we conclude that Pk(x,Pj) and Pk(x,P~l) are null polynomials, hence
Pki (Pj) = Pki (Pj-1) = 0.
It follows that polynomials P^(x) are divisible by xn — 1 and we immediately conclude that P^(x) are also null polynomials. Therefore, we have Pk(x1,x2) = 0. Injectivitv is proved. Finally, it remains to show that
(n+Si r2+ S2
© © *,j
=1 =1
Using (5), we obtain dimFq (^¡=+si s2 Bi,j)
ri ri+si
1
= V2 I 4£(m) + 2 dimFq (M2(Fg[ai + a-1])) + 2 ^ dimFq (M2(Fg[ai])) I +
\ i=£(m)+1 i=ri+1 J
(r-i+si n+si \
2{(m)dimFq M2(Fq)+ ^ dimFq (M4(Fg[ ai + a-1]) + ^ dimFq (M^[ai])) I
= i +1 = i +1
ri+si
(n — 2 S2) (4C(m)+4 ^ deg fj + 8 ^ deg f3 | +
i=£,(m)+1 i=ri+1
i i+ i
An" 13
+s2 |8{(m)+8 deg/j + 16 ^ deg/j | = 4(n — 2s2)m + 8s2m = 4mn
\ i=£(m)+1 i=ri + 1 J
= dimFq FqQm,n.
Hence p is an isomorphism. □
Lemma 5. Let R be an algebra with identity 1r and (1r + 1r) has an inverse element in R; then R(h)2 - R ® R and R((hi )2 x (h2)2) - R ®R ®R ®R.
Доказательство. Indeed, the map <p : R(h)2 ^ R ®R such that
p(ri + ^h) = (ri + Г2, ri _ Г2)
is an isomorphism and
V (r i, ^2) =
T\ + Г2
+
Г\ _ Г2
h.
(1r + 1r ) (1r + 1r) Since R((hi)2 x (h2)2) — (R(hi)2) (h2)2 it follows that
R((hi)2 x (h2)2) - R(h)2 ® R(h)2 - R ®R ®R ® R.
□
Now we can establish the Wedderburn decomposition of FqQmifl in the case gcd(2mn, q) = 1 n \ - 1
Teopema 2. Let gcd(2mn, q) = 1 andn \ q—1; the nFqQmifl has the Wedderburn decomposition of the form:
d : FqQ.
q^im,n
ri+Sl T2+S2 i=i j=i
1
(12)
Fg ® Fg ® Fg ® Fg,
(1 < < ( m)) Л (1 < < 2)
M2(Fg [ Qi + Q-i]) ® M2(Fg Q + Q-i]), (£(m) + 1 < i < r 1) Л (1 <j< Г2)
— <
( n + 1 < i < r 1 + si) Л (1 < j < Г2) (1 < < ( m)) Л ( 2 + 1 < < 2 + 2)
(£(m) + 1 < i < r 1) Л (Г2 + 1 < j < Г2 + s2) ( r\ + 1 < г < r 1 + Si) Л ( Г2 + 1 < j < Г2 + s 2)
M2(Fg[ Qi]) ® M2(Fg[ Qi]) ,
M2(Fg) ® M2(Fg),
M4(Fg [ Qt + Q- i]), ^M4(Fg [ Qi]),
Доказательство. Let us define the maps nj
1 < < ( m) 1 < < 2
nj : Fg((hi)2 x (h2)2) ^ F4
nj(Xo + Xhi + X2h2 + X3hih2) = (Po + P + P2 + P3, Po + P _ P2 _ P3,
Po _ P + P2 _ P3, Po _ P _ P2 + P3);
( m) + 1 < < 1 < < 2
: M2(Fg [Qi + Q-i])(h)2 ^ M2(Fg [Qi + Q-i]) ® M2(Fq [Qi + Q-i]),
ThJ(Xo + Xh) = (Xo + X, Xo _ X);
3. for п + 1 < i < r 1 + si and 1 < j < r2:
: M2(F,[Qi])(h)2 ^ M2(F,[Qi]) ® M2(F,[Qi]), (Xo + Xh) = (Xo + X, Xo _ X);
4. for 1 < i < £(m) and r2 + 1 < j < r2 + s 2
: M2(F,) (h)2 ^ M2 (F,) ® M2 (F,), Titj (Xo + Xh) = (Xo + X, Xo _ X);
Using lemma 5 we conclude that r^j a re Fg-algebras isomorphisms. Now let
r 1+ S1 Г2 + S2
d ^ Ф di'j, i=i j=i
di,j :=
^i^ij,
(1 <i< i(m)) Л (1 <j< Г2) (£(m) + 1 < i < n) Л (1 < j < r2) (n + 1 < i < n + si) Л (1 < j < Г2) (1 <i< £(m)) Л ( Г2 + 1 <j< Г2 + s 2) (£( m) + 1 <i< п) Л ( r 2 + 1 <j< Г2 + s 2) (ri + 1 <i < ri + Si) Л (Г2 + 1 < j < г 2 + s 2)
Therefore, using theorem 1 we see that d is ад isomorphism. □
Замечание 2. Since Qm,n ~ Qn,m, we can also use these theorems in the case n { q — 1 but m \ q — 1.
3. Primitive central idempotents of F qQm^n
Let R be a ring. Recall, i E R is an idempotent if i2 = i. Two idempotents i\, i2 E R are called orthogonal if i\i2 = i2i\ = 0. An idempotent i is called central if ri = ir for all r E R. An (central) idempotent i is said to be primitive (central) idempotent if i cannot be written as i = i' + i" where i' and i" ^re such (central) idempotents that i', i" = 0 and i'i" = 0.
In this section, firstly, we consider the set of idempotents of cyclic group algebra. This set allows us to explicitly construct p-1 and d-1, where isomorphisms p and d are defined in the section 3. Then we use d-1 ^o describe the primitive central idempotents of FQm,n in the case gcd(2mn, q) = 1, n \ q — 1. Note that the maps p"^d d-1 could also be useful to study the algebraic structure of group codes over Qm,n.
Let gcd(fc, q) = 1. Let ~Rk := Fg[x]/(xk — 1), where (xk — 1) the principal ideal of Fq[x]
generated by xk — 1. ft is known (see [18], lemma 2.1) that for monic polynomial g(x) \ xk — 1 an element
k (x):= — Mx)*)T xk — 1
к
g(x)
(13)
is the principal idempotent of the ideal ~Rk \^^^y], where \g(x)] E ~Rk is the equivalence class of ( x)
Lemma 6. Let g(x) be a monic irreducible divisor of xk — 1 and a be a root of g; then
(i) ek(a) = 1;
XK — 1
(ii) e'k(f3) = 0 for any root ¡3 of the polynomial Xg-X) .
Доказательство. The definition (13) yields (ii). The Chinese reminder theorem implies that the map
Ф
F [x]
F [x]
(g(x))^ ((xk — 1)/g(x)):
P(x) ^ ( P(x) mod g(x), P(x) mod
is an isomorphism. Since e^(x) is ад idempotent and (FX) is a field, it follows that
xk 1
g(x)
X)
e'k(x) mod g(x) = 1.
Hence e'(a) = 1. □ It is well known that Fg(h)k ~ hence for any g(x) \ (xk — 1) we have e'k(h) E Fg(h)k is an idempotent.
Let S be a set. By ids we denote the identity map on S.
Ti,j Ui, j
Ti.jO'il'iJ
Ti,j Ui, j
Lemma 7. Let gcd( q, mn) = ' 1 < г < £(m) and 1 < j < r2. Let
Vij : Fq((hi)2 x (h2)2) ^ FqQmn
be a map defined by
(Po +Pihi + P2h2 + P3hih2) := (po +Pib + P2C + с) em(ai)e^.(02). Tften = idim(jy.,j) and Vi'j'^ij = 0 if г' = г оr j' = j.
Доказательство. Lemma 6 implies that Vi'j'p,i,j = 0 if i' = i о г j' = j. We have
({po + Piь + P2C + P3bc) ermi(ai)e^.(a2)) = po + Pihi + Р2Л.2 + Р3Л.3.
Hence = idim(^)• □
Lemmas 8-12 are proved in the same way. Recall, the maps Ui,j are Fg-algebras homomorphism and their images in fact were described in (7)—(10).
Lemma 8. Let £(m) + 1 < i < r\ and 1 < j < r2. Let
¡¿i,j • im(uij)
: С
Po(Qi) Pi(Q- )\ , fp2(Qi) Р3(ох- )
)+( i
Pi(Qi) Po(<Qi )J \P3(<Qi) P2(Qi )
h
po(a\) + bp2(a\) + cpi (ai) +
+bcp3(ai) ermi(ai)e^.(02).
Then = idim(^i,j) and Vi'fHij = 0 if i' = i оr j' = j.
Lemma 9. Let r 1 + 1 <i <r 1 + si and 1 < j < r2. Let
^ FgQm,n
: С
Po(Qi) P2(<Q- )\ , fp4(Qi) Рб(ох- )
1,3 ' \Pi(Qi) P3(qq~l)) XPb(QQi) Pt^-1 ))
Po(ai) + bpi(ai) + cpi (ai) +
+bcpb(al) ermi(ai)eng.(a2) +
+
P2(ai) + bp&(ai) + cp3(a\) + bcp7(ai) em*(ai)eng (a2).
Then = idim(^m>j) and n'fHij = 0 if г' = г оr j' = j.
1 < < ( m) 2 + 1 < < 2 + 2
: im() ^ FqQm,n,
Hj : S) + Ь + bpi+cpi + bCP5]em(ax)е™ Ы +
+
P2 + bp3 + фб + bcpr em (ai)еП* (a2)
Tften = idim(^i,j) and n'fHij = 0 if г' = г оr j' = j.
Lemma 11. Let £(m) + 1 <г< r1 and r2 + 1 < j < r2 + s2. Let
: im(uij)
A
i,3 ■
fpo(ai) Pi(ai) P2 (a'1) Pe(a-~1)\
P\(ai) Pb(ai) рз ( a'1) p7(a'1)
P2(ai) Pe(ai) po (a-1) p^a'1)
\p3(ai) P7(ai) pi (a-1) p5(a~1)J
Po(a1 )+bp1(ax)+cp2(ax)+bcp3(ц) e™( ах)e^(02) + + Ръ(а\) + bpi(a1) + cp7(a\) + Ьсрб(а\) e™(01)еП*(02).
Then UitjAi,j = idimCi/i,^) and "i'j'Vij = 0 if г' = г оr j1 = j.
1 + 1 < < 1 + 1 2 + 1 < < 2 + 2 Ai,j : im(г/ij) ^ ¥qQm,n,
Ai,3 :
(Po(ai) P4(ai) p8(a-1) p^a'1^
P1(ai) Pb(ai) pg(a--1) p^a'1)
P2(ai) pe(ai) p^a'1) pu(a'1)
\p 3(ai) p7(ai) pn( a'1) p^a"1)/
^ po(a1) + bp1(a1) + cp2(a1) + 6cps(a1) em(a1)e^(a2) + + p5(a1) + bp4(a1) + cp7(a\) + bcpe(a1) e™( a1)e1^*( a2) + + p1o(a1) +bp11(a1) + cp8(a1) + pg^) ej*(a1)еП((12) +
+ p15(a1)+ 6p14(a1)+cp13(a1)+ 6cp12(a1) ej*(ц)e^(a2).
Tften Vi,jTi,j = idim() and Vi>j>Ai,j = 0 if i1 = i оr j1 = j.
In the following theorem bv the use of these lemmas we describe p'1 and d Теорема 3. (i) Let gcd(mn, q) = 1 and let
1
T1+S1T2+S2 Г-1+ S1T2+S2
EE Рч : © Ф Bi ^ ^ FQ
i=1 3=1
i=1 3=1
1
1
Pi,3 := S Ai,3Gj
Then p'1 = p.
(ii) Let gcd(2mn, q) = 1 and let
(i(m) + 1 <i< r1) Л (1 <j< r2)
(£(m) + 1 <i< п) Л (Г2 + 1 <j< Г2 + s 2)
otherwise
ri+ S1T2+ S2 T1+S1T2+ S2
d = E E di,3 : © © Aj ^ =1 =1 =1 =1
di,j :=
PiJ^i1 T-jS -1
(1 <i< £(m)) Л (1 <j< r2) (£( m) + 1 < г < n) Л (1 < j < r2) (n + 1 < i < n + si) Л (1 < j < Г2) (1 < < ( m)) Л ( 2 + 1 < < 2 + 2) (£( m) + 1 < i < n) Л (Г2 + 1 < j < Г2 + s 2) (Г1 + 1 < i < П + Si) Л (Г2 + 1 < j < Г2 + s 2)
Tften d-1 = d.
Доказательство. Since p is of the form (6), directly computing from lemmas 8-12 we obtain
ri+Si T2+S2 ri+Si T2+S2 ri+Si T2+S2 ri +S1 T2+S2
pp= ©©EE Pi>3pi'3' = Ф Ф Pi'lPiJ = Ф Ф id^ i=1 j=1 i'=1 j'=1 i=1 i=1 i=1 i=1
Since p is ад isomorphism (see theorem 1), it follows that p-1 = p. This is also true in the case (ii). □
Now we can describe the primitive central idempotents in FqQm,n in the case gcd(2mn, q) = 1 and n \ q — 1.
Теорема 4. Let gcd(2mn, q) = 1 and n \ q — 1; iften FqQm,n has
1) 4{(m)r2 primitive central idempotents of the form
+ + + 4
em («1) еП, («2),
+ — —
emm («1)е ' («2),
— + — F"
emm («1) e™. («2)
— — +
emm («1)е n ( «2)
1 < < ( m) 1 < < 2 2(r 1 — £( m))r2 primitive central idempotents of the form:
e — b e + b
етмк(«2), -^-¿mM^(«2),
2
( m) + 1 < < 1 1 < < 2
2 1 2
( «1) + em* Ы) en, («2) ^ (e5m(«1) + ^* ( «1)) e' ( «2)
2 V Ji Ji ) gj 2
1 + 1 < < 1 + 1 1 < < 2 2 ( m) 2
emm( «1) (e',. ( «2) + e'* ( a*)) , ^е?*(«1) (e' ( «2) + e'* («2))
2 "Ji^-"1' \~yj\~2/ ■ -д^-2/^ 2 ^fi 1 < < ( m) 2 + 1 < < 2 + 2
( 1 — ( m)) 2
em(«1) (en,(«2) + e'* («2)) ,
( m) + 1 < < 1 2 + 1 < < 2 + 2
6) s\s2 primitive central idempotents of the form,:
(€( ai)+ em* (ai)) (e^ Ы + ^ Ы) ,
where r\ + 1 < i < ri + Si and r2 + 1 < j < r2 + s2. Доказательство. Let ЛЬеа semisimple ring and let
f : R ^ Ri ®R2 ф-^Ri,
where Ri are matrix rings over division rings, be the Wedderburn decomposition of R. It is well-
R
ei = f'1(0 ф-^ф 0 ф Ii ф 0 ф-^ф 0),
where Ii £ R is the identity element.
Therefore, using theorems 2 and 3, we obtain l)-6). Indeed, consider for example 1) in the case i = 1 and j = 1. Using theorem 3 we get
d'1 ((1, 0, 0, 0) ф 0 ф 0 •••ф 0) = A1,14,1(1, 0, 0, 0).
By definitions of т1,1 from theorem 2 and am from theorem 3 we obtain
~1n n n n\ f1 + h1 + h2 + h1hA e + b + c + bc A1,1TU(1,0, 0, 0) =Ац[ -4- ) =-4-ejm (a1)e^ (a2).
4 ) 4
Similarly we can evaluate
d-1 ((0,1,0,0) © 0 © 0 ■■■© 0) ...d-1 ((0, 0,0,1) © 0 © 0 ■■■© 0)
and remaining primitive central idempotents for 1 < i < £(m) and 1 < j < r2.
Now let's consider 2). In the case £(m) + 1 < i < r\ and 1 < j < r2 using definitions of d and from theorem 3 and Ti,j from theorem 2 we get
d-1(0 ©■■■© 0 © (E © 0) © 0 ©■■■© 0) = ^o^t— (E © 0) = ^ (E + 0h) = ef. (ai) eng] (02) and
+
d-1(0 © ■ ■ ■ ©0 © (0 © E) ©0 © ■ ■ ■ ©0) = (0 ©E) = ^(0 + Eh) = -yef.(ai)eng.(02),
here E denotes the identity matrix and (0 ф E), (E ф 0) £ Ai,j.
The remaining cases 3)-6) are proved in the same way. For example, let's consider 6). In the case r1 + 1 < i < r1 + s1 and r2 + 1 < j < r2 + s2 we have
d_1(0 ф^ф 0 фЕ ф 0 ф^ф 0) =Ai,j (E) = (ej1. (ц) + ej (ц)) (e™ (a2) + eg Ы) , here E £ is identity mat rix. □
Замечание 3. Note that, FqQm,g splits into internal direct sum of minimal two-sided ideals Ik С FqQm,n. Each Ik is isomorphic to one of the simple direct summands in (12) and generated by an idem,potent from theorem 4-
4. An application to algebraic codding theory
Now we can establish the structure of the group codes over Qr, notation. Define
i. First, let us introduce the
M(i, j) := <( 4,
and
Fi : =
(1 <i< £(m)) Л (1 <j< r2),
(ri + 1 < г < Г\ + si) Л (r2 + 1 < j < Г2 + s 2),
otherwise
X, 1 < г < £(n),
Fç [ ai + a"1], £(n) + 1 <i< r,
[ ai], r + 1 < i < r + s.
Let k G N, let F be a field and V be a subspace of Ffc^v Z(F, k, V) we denote the set of all matrices K G Mfc(F) such that Kv = 0 fe a 11 v G V.
In [21], p. 93, it was proved that any left ideal of M&(F) is of the form 1 (F, k, V) and there is one-to-one correspondence between the left ideals of M k(F) and the linear subspaces of Ffc.
Teopema 5. Let gcd(2 mn, q) = 1 and n | (q — 1). For any group code C C FqQm,n there exist subspaces Vijk C Fi.
M ( г ,j)
such that
ri+s 1 Г2+ S2
d(°) = © ® Сг,3, C:
г=1 j=1
,
(14)
et=i ш, 1, Уг,3,к ), (i <i< am)) л (i <j< r2)
1 ( Fz, 2, Viji) ®1 (Fi, 2, VhJ,2), (t(m) + 1 < i < п) Л (i <j< Г2)
X( Fi, 2, Vij,i) Ф1 (Fi, 2, Vi,j,2), (r 1 + 1 < г < n + Si) Л (1 <j< Г2)
1 ( Fi, 2, Vij,i) ф1 (Fi, 2, Vi,j,2), (1 < г < C(m)) Л (г2 + 1 < j < Г2 + S2)
1 ( Fi, 4, Vij, 1), (C(m) + 1 < i < n) Л (Г2 + 1 < j < Г2 + S2)
( Fi, 4, Vij,i), (r 1 + 1 < i < n + si) Л (г2 + 1 < j < Г2 + S2)
Contrariwise, for any subspaces Vi,j,k С F^(iйе sei
r+s
d-i( © Ci,,), i,j=i
(15)
with Citj defined in (15), is a group code in FqQm,n. Доказательство. Consider (12) from theorem 2. Let
r-1+Si Г2+ S2
A^ ©Я
г=1 3=1
,
Since d : FqQ.
A is an isomorphism, it follows that there is one-to-one correspondence between the codes in FqQm,n and the left ideals of A. It is well-known that any left ideal of a direct sum of algebras is a direct sum of left ideals of summands. It is established that the ideals of summands are of the form 1(Fi, t, Vi,j,k). Hence the theorem is entirely proved. □
Consider a group code C C FqQm,n. We obliviously have the length of C equals to 4mn and the dimension can be evaluated by following formula:
r-1+Sl T2+S2
dim(C) = £ £ dim(Cij).
i=1 3=1
122
K. B. Bc/u-iici;. B. M. , Icyii.iMK
5. Conclusion
In the paper we considered the bidihedral group Qm,n and its group algebra FqQm,n. In the case gcd( mn, q) = 1 and n | q — 1 we obtained the structural theorem for FqQm,n. Then we used it to explicitly describe the Wedderburn decomposition of FqQm,n in the case gcd(2mn, q) = 1. Moreover, we constructed inverse isomorphisms p-1 and d-1, which helped us to describe the central primitive idempotents of FqQm,n.
Finally, we used the Wedderburn decomposition d and d-1 to algebraically describe all codes in FqQm,n in the case gcd(2mn, q) = 1 and n | (q — 1). In addition, it is easy to find their length and dimension.
Further research is needed to find among these codes promising classes of error-correcting codes with good parameters and to construct decoders.
REFERENCES
1. Milies, C. P. k Sehgal, S. K. 2002, An inroduction to Group Rings, Kluwer Academic Publishers, Boston.
2. Lang, S., 2002, Algebra, Springer-Verlag, New York.
3. Kelarev, A.V. k Solé, P. 2001, "Error correcting codes as ideals in group rings", Contemp. Math., vol. 273, pp. 11-18.
4. Kouselo, E., Gonsales, S., Markov, V.T., Martines, K. k Nechaev, A.A. 2012, "Ideal representations of Reed-Solomon and Reed-Muller codes", Algebra Logic, vol. 51, no. 3, pp. 195-212.
5. Berman, S.D. 1967, "On the theory of group codes", Cybernetics, vol. 3, pp. 25-31.
6. Charpin, P. 1983, "The Extended Reed-Solomon Codes Considered as Ideals or a Modular Algebra" North-Holland Mathematics Studies, vol. 75, pp. 171-176.
7. Tumavkin, I. N. 2018, "Group Ring Ideals Related to Reed-Muller Codes", J Math Sei, vol. 233, pp. 745-748.
8. Zimmermann, K.-H. 1994, Beitrage zur algebraischen Codierungstheorie mittels modularer Darstellungstheorie, Bayreuther Mathematische Schriften Vol. 48, University of Bayreuth.
9. Assuena, S. k Milies, C.P 2019, "Good codes from metacyclic groups", Contemp. Math., vol. 727, pp. 39-49.
10. Olteanu, G. k Van Gelder, I. 2015, "Construction of minimal non-abelian left group codes", Des. Codes Crvptogr., vol. 75, no. 3, pp. 359-373.
11. Vedenev, K.V. k Deundvak, V. M 2018, "Codes in Dihedral Group Algebra" (in Russian), Modeling and Analysis of Information Systems, vol. 25, no. 2, pp. 232-245.
12. https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/Post-Quantum-Cryptography-Standardization Last visited 1.07.2019.
13. Minder, L. k Shokrollahi, A. 2007, "Cryptanalvsis of the Sidelnikov cryptosvstem", Lecture Notes in Computer Science, vol. 4515, pp. 347-360.
14. Chizhov, I. I. k Borodin, M. A. 2014, "Effective attack on the McEliece cryptosvstem based on Reed-Muller codes", Discrete Mathematics and Applications, vol. 24, issue 5, pp. 273-280.
15. Sidelnikov, V. M., k Shestakov, S. O. 1992, "On an encoding system constructed on the basis of generalized Reed-Solomon codes",Discrete Mathematics and Applications, vol. 2, issue 4, pp. 439-444.
16. Broche, O. k Del RiO, A. 2007, "Wedderburn decomposition of finite group algebras", Finite Fields and Their Applications, vol. 13(1), pp. 71-79.
17. Bakshi, G.K., Gupta, S., k Passi, I. B.S. 2013, "The structure of finite semisimple metacvclic group algebras", J. Ramanujan Math. Soc, vol. 28(2), pp. 141-158.
18. Martinez, F.B. 2015, "Structure of finite dihedral group algebra", Finite Fields and Their Applications, vol. 35, pp. 204-214.
19. Coxeter, H.S., k Moser, W. O. 2013, Generators and relations for discrete groups, Springer Science k Business Media.
20. Magnus, WT., Karrass, A., k Solitar, D. 2004, Combinatorial group theory: Presentations of groups in terms of generators and relations, Courier Corporation.
21. Jacobson, N. 1956, Structure of rings, Vol. 37, American Mathematical Soc.
Получено 7.08.2019 г. Принято в печать 12.11.2019 г.
