FIFTH LECTURE
6.2 Metacyclic Codes and Equivalence Questions
below.
e0 e1 e2 dimFqC w(C)
bbCc2 C\ pm1 1
\ Cpm2 2
1 4pm11pm22 (1−bb)Cc2 C\
pm1 1
\ Cpm2 2
1 4pm11pm22 bb(1−Cc2) C\
pm1 1
\ Cpm2 2
1 4pm11pm22 (1−bb)(1−Cc2) C\
pm1 1
C\ pm2
2
1 4pm11pm22
Cc2 C\
pm1 1
\ C
pm2−j 2
−C \ pm2−j+1
2
2φ(pj2) 4pm11pm22−j
Cc2 C\
pm1−i 1
−C \ pm1−i+1
1
\ Cpm2 2
2φ(pj1) 4pm11−ipm22
(1−Cc2) C\
pm1 1
\ Cpm2−j
2
−C \ pm2−j+1
2
2φ(pj2) 4pm11pm22−j (1−Cc2) C\
pm1−i 1
−C \ pm1−i+1
1
\ Cpm2 2
2φ(pj1) 4pm11−ipm22
Cc2 C\
pm1−i 1
−C \ pm1−i+1
1
\ C
pm2−j 2
−C \ pm2−j+1
2
2φ(pj1)2φ(pj2) 8pm11−ipm22−j (1−Cc2) C\
pm1−i 1
−C \ pm1−i+1
1
C\ pm2−j
2
−C \ pm2−j+1
2
2φ(pj1)2φ(pj2) 8pm11−ipm22−j
Similar results were obtained by Dutra [23, Cap´ıtulos 4 e 5] for group codes over the quaternion groups.
For G a metacyclic finite group such that gcd(q,|G|) = 1, Sabin and Lomonaco [63], by using group representation theory, proved that codes generated by central idempotents in FqG are combinatorially equivalent to abelian codes. This motivated the search for left minimal codes in FqG.
Considering the group algebra of a non-abelian split metacyclic group G over a finite field Fq, Assuena [3] in his Ph.D. thesis, found a necessary condition under whichFqGhas the minimum number of simple components.
Theorem 6.8. [3, Teorema 2.1.16] Let G be a metacyclic group and Fq a finite field with q elements such thatgcd(q,|G|) = 1. If the number of simple components of the group algebra FqG is minimal, then U(Zn) = hqi¯ and U(Zm) =h¯iih¯qi.
In his thesis, Assuena used the structure of the group to determine the minimal metacyclic codes for a non abelian split metacyclic group of order pm`n, with p and ` odd prime numbers, under the conditions that FqG is semisimple and the number of simple components of FqGis minimum.
ForDpm, the dihedral group of order 2pm, and Fq a finite field such that gcd(q,2pm) = 1, he constructs left minimal codes that are not combina-torially equivalent to abelian codes and also exhibits one case where a left minimal code is more efficient then the abelian ones of the same length, giving a positive answer to a conjecture of Sabin and Lomonaco [63].
Further studies on group codes are given in [11], where it is defined a (left) G-code as any linear (left) code of length n over a field Fq which is the image of a (left) ideal of a group algebra via an isomorphismFqG−→Fnq
which maps the finite group Gof order n to the standard basis ofFnq. Their ideas are used in [64] to study two-sided and abelian group ring codes and in [30], where Garc´ıa Pillado et al. first communicated an example of a non-abelian S4-code over F5. The full proof of this computacional construction was given later in [31]. New examples of non-abelianG-codes are given in [32]
and, particularly, using the group SL(2;F3) instead of the symmetric group, they prove, without using a computer for it, that there is a code over F2 of length 24, dimension 6 and minimal weight 10. This code has greater minimum distance than any abelian group code having the same length and dimension overF2, and, moreover, it has the greatest minimum weight among all binary linear codes with the same length and dimension.
In [24] Elia and Garc´ıa Pillado give an overview of the properties of ideal group codes defined as principal ideals in the group algebra of a finite group
G over a finite field Fq and present their encoding and syndrome decoding.
They also describe in detail a correction of a single error, using syndromes.
7 Codes over rings
In the 1990’s many papers on cyclic codes over rings started to appear, motivated by the fact that good non linear binary codes were related to linear codes over Z4 (see, for example, [17, 38, 55]). The paper [36] by Hammons et al. was even the best paper award for Information Theory of the IEEE-IT Society in the 1996 Symposium of IT - Whistler (Canad´a).
Wood [77] addressed the problem of duality for modules over finite chain rings and applied it to equivalence of codes and to the extension theorem of MacWilliams.
In [14] Carlderbank e Sloane determine the structure of cyclic codes over Zpm. Later on, in [38] Kanwar and L´opez-Permouth did the same, but with different proofs. With the same techniques, Wan [74] extended the results from [38] to cyclic codes over Galois rings. Em 1999, Norton and S˘al˘ agean-Mandache in [53] extended results of [14, 38] to cyclic codes over finite chain rings and later on, in 2004, Dinh and L´opez-Permouth in [19] prove the same results in a different way.
Codes over rings developed even more in the beginning of the 21st century that they deserved a CIMPA Summer School in 2008 [72]. Further works can be found in [18], [42], [48]. A small survey on the subject is [33].
In his thesis [69, 70], Silva used group ring approach to characterize cyclic codes over chain rings, their duals and some conditions on self-dual codes, simplifying the proofs and improving results given in [19].
LetR be a finite commutative chain ring with unity such that |R|=qk, for a prime q. For M the maximal ideal of R, the quotient R = MR is a field and we work under the hypotesis that q -| G |, for a finite cyclic group G.
Under these conditions, the group ring RG is a principal ideal ring, as Silva proves in [69, Teorema 2.1.9], after characterizing all the ideals in RG. The following general fact is a basis for all this work.
Theorem 7.1. [69, Teorema 2.1.2]Let R be a local ring, with maximal ideal M =< a > and | R |=qk, and G a cyclic group of order n such that q - n.
If {e0, ..., em} is a full set of primitive orthogonal idempotents in RG, then {e0, ..., em} is a full set of primitive orthogonal idempotents in RG.
The next theorem characterizes all cyclic codes of lengthn over the local ring RGei (see [69, Corollary 11.31]), for R a chain ring and ei a primitive orthogonal idempotent, translating results of [19] to the group ring setting.
To simplify the notation we write (RG)ajei ashajeii.
Theorem 7.2. [69, Teorema 2.1.3] Let R be a commutative finite chain ring with unity, |R |=qk, M =hai the maximal ideal of R and t the nilpotency index o ´ındice de nilpotˆencia of a in R. Let G =Cn such that q -n. If I is an ideal of RGei, then I is of the form I =
akiei
, with 0≤ki ≤t.
Corollary 7.3. [69, Corollary 2.1.4] Under the same hypothesis of Theo-rem 7.2, the ideal RGei is indecomposable in RG and the code hat−1eii is minimal.
From this we have a characterization of all cyclic codes of length n over chain rings.
Theorem 7.4. Let R be a commutative finite chain ring with unity, | R|=
qk, M =haithe maximal ideal of R andt the nilpotency index ofa inR. Let G=hg0/ g0n= 1i be such that q -n and {e0, ..., em} be a full set of primitive orthogonal idempotents of RG. Then:
(i) [69, Teorema 2.1.5] If I is an ideal of RG, then I is of the form I =I0⊕...⊕Im, with Ii =
akiei
, for 0≤ki ≤t.
(ii) [69, Teorema 2.1.8] The number of such codes of length n over R is (t+ 1)m+1.
One important data in a code is its number of words. Next theorem gives this number for cyclic codes over finite chain rings. We have
RG=RGe0⊕...⊕RGem ' R[x]
hxn−1i ' R[x]
hf0i ⊕...⊕ R[x]
hfmi,
where fi are irreducible factors of xn−1 and , after reordering the indexes if necessary, we have RGei ' R[x]hf
ii. Hence,|RGei |=|R |wi, for wi =deg(fi).
Theorem 7.5. [69, Teorema 2.1.7] Under the same hypothesis of Theo-rem 7.2, let C be a cyclic code of the form C =
aki1ei1
⊕...⊕
akireir
in RG. The the number of words in C is |C |=|R |
r
X
s=1
(t−kis)wis .
Considering ∗ : RG −→ RG the classical involution, Silva also gives a description of the dual cyclic codes in RG as follows.
Theorem 7.6. [69, Teorema 2.2.3] Under the same hypothesis of Theo-rem 7.4, the dual code of a cyclic code C =
ak0e0
⊕...⊕
akmem , with 0≤ki ≤t, is C⊥=⊕Pm
r=0
at−krer∗ .
As in [19], Silva in [69, Section 2.2] states the conditions for the ring R under which the group ring RG admits self-dual codes.
Chapter 3 of [69] is dedicated to codes over chain rings of length pn, for a prime p, extending the results of Ferraz and Milies [28] and of Melo [49]
to this context. Silva also proves in [69, Teorema 3.0.14] some facts about the size of such codes and computes minimum weight of these codes [69, Teoremas 3.0.15 to 3.0.18], similarly to Theorem 5.14. He also discusses about free codes in RG in [69, Section 3.1] and about MDS codes of length pn over R in [69, Section 3.2]. Finally, in [69, Chapter 4], Silva proves all such results for cyclic codes of length 2pn over finite chain rings.
There are also interesting discussion on equivalence of linear codes over rings in [20, 21, 76, 78].
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