Binary abelian codes of exponent pq An abelian code is a proper

31 Pages·2011·0.3 MB·English

Binary abelian codes of exponent pq
Gladys Chalom ∗
Raul Antonio Ferraz †
Marinˆes Guerreiro‡
Francisco C´esar Polcino Milies§
July 12, 2011
An abelian code is a proper ideal in a group al-
gebra of an abelian group. We will consider min-
imal abelian codes in the group algebra IF G,
2
for G an abelian group of exponent p · q, where
p and q are distinct odd primes satisfying the
following conditions:
(i) gcd(p − 1, q − 1) = 2 and (1)
¯ Z Z
(ii) 2 generates U( ) and U( ) (2)
p q
∗Supported by PROCAD/CAPES (Brazil)
†Supported by PROCAD/CAPES (Brazil)
‡Supported by FAPEMIG APQ CEX 00438/2008 and FAPESP (Brazil)
§Supported by Projeto Tem´atico FAPESP (Brazil)
1
1 Basic Facts
Lemma 1.1. Let p be a positive prime num-
N∗
ber and r, s ∈ . Then
∼
F F F
⊗ = gcd(r, s) · .
r F s
p p lcm(r,s)
p p
Remark 1.2. Notice that any extension L of
F
of even degree contains a subfield K with
2
four elements, hence there exists an element
3
1 (cid:54)= a ∈ L such that a = 1.
2
N
Lemma 1.3. Let r, s ∈ be non-zero ele-
ments such that gcd(r, s) = 2. Let
F F
1 (cid:54)= u ∈ and 1 (cid:54)= v ∈ be elements
r s
2 2
3
satisfying the equation x = 1. Then
∼
F F F F
⊗ = ⊕ (3)
r F s rs rs
2 2
2
2 2 2 2
2 2
and e = (u ⊗ v) + (u ⊗ v )
1
2 2
and e = (u ⊗ v ) + (u ⊗ v) are the primi-
2
tive idempotents corresponding to the simple
components of (3).
3
F
2 Codes in (C × C )
2 p q
pq
For p (cid:54)= q odd primes, G = (cid:104)g | g = 1(cid:105),
q p
denote a = g , b = g and write G = C × C .
p q
(cid:88)
ˆ
For a subgroup H ≤ G, we set H = h
h∈H
and for an element x ∈ G, we set xˆ = (cid:104)(cid:99)x(cid:105).
4
Theorem 2.1. Let G = (cid:104)a(cid:105)×(cid:104)b(cid:105) be as above
and assume that p and q satisfy (1). Then
F
the primitive idempotents of G are:
ˆ ˆ
e = G, e = aˆ(1 − b), e = (1 − aˆ)b,
(cid:98)
0 1 2
2 2 2 2
e = uv + u v and e = uv + u v,
3 4
where

0 2 4 p−3
2 2 2 2
a + a + a + · · · + a ,




if p ≡ 1(mod 4) or
u =
0 2 4 p−3
2 2 2 2
1 + a + a + a + · · · + a ,



 if p ≡ 3(mod 4)
(4)
and

0 2 4 q−3
2 2 2 2
b + b + b + · · · + b ,




if q ≡ 1(mod 4) or
v =
0 2 4 q−3
2 2 2 2
1 + b + b + b + · · · + b ,



 if q ≡ 3(mod 4).
(5)
5
F
3 Abelian codes of (C × C )
2 p2 q2
Recall some facts on primitive idempotents in
group algebras for abelian p-groups according
to [FP].
Let A be an abelian p-group. For each sub-
group H of A such that A/H (cid:54)= 1 is cyclic,
∗
consider the only subgroup H of A contain-
∗
ing H such that |H /H| = p and define e =
H
ˆ ˆ∗
H − H . Clearly e (cid:54)= 0 and we have the
H
following:
Lemma 3.1. (Lemma 5, p. 389 [FP]) The
elements e , defined as above together with
H
ˆ
e = A form a set of pairwise orthogonal
A
idempotents of F A whose sum is equal to 1.
6
Let p and q be distinct odd prime numbers
such that
gcd{p − 1, q − 1} = 2, gcd{p, q − 1} = 1,
gcd{p − 1, q} = 1 and
¯ Z Z
2 generates U( ) and U( ). (6)
2 2
p q
Theorem 3.2. Let p and q satisfy (7) and
G = (cid:104)a(cid:105) × (cid:104)b(cid:105) be an abelian group of type
C × C , where C = (cid:104)a(cid:105) and C = (cid:104)b(cid:105).
2 2 2 2
p q p q
F
Then the minimal codes of (C × C ) are
2 2 2
p q
described in the following table.
7
Ideal Primitive Dimension Code Remarks
Idempotent Weight
I a(cid:98)b 1 p2q2
0 (cid:98)
I a((cid:98)b+b(cid:98)q) q −1 2p2q
1 (cid:98)
I2 ((cid:98)a+a(cid:98)p)(cid:98)b p−1 2pq2
I a(1+b(cid:98)q) q(q −1) 2p2
3 (cid:98)
I4 (1+a(cid:98)p)(cid:98)b p(p−1) 2q2
I uv +u2v2 (p−1)(q−1) ?? u = ap(a20 +a22 +···
5 2 (cid:98)
+a2p−3) ifp ≡ 1(mod 4)
u = ap(1+a20 +a22 +···
(cid:98)
+a2p−3) ifp ≡ 3(mod 4)
I uv2 +u2v (p−1)(q−1) ?? v = b(cid:98)q(b20 +b22 +···+b2q−3) ifq ≡ 1(mod 4)
6 2
v = b(cid:98)q(1+b20 +b22 +···+b2q−3) ifq ≡ 3(mod 4)
I uv +u2v2 (p−1)q(q−1) ?? u = ap(a20 +a22 +···+a2p−3) ifp ≡ 1(mod 4)
7 2 (cid:98)
u = ap(1+a20 +a22 +···+a2p−3) ifp ≡ 3(mod 4)
(cid:98)
I uv2 +u2v (p−1)q(q−1) ?? v = (b20q +b22q +···+b2q−3q) ifq ≡ 1(mod 4)
8 2
v = (1+b20q +b22q +···+b2q−3q) ifq ≡ 3(mod 4)
I uv +u2v2 p(p−1)(q−1) ?? u = (a20p +a22p +···+a2p−3p) ifp ≡ 1(mod 4)
9 2
u = (1+a20q +a22q +···+a2p−3q) ifp ≡ 3(mod 4)
I uv2 +u2v p(p−1)(q−1) ?? v = b(cid:98)q(b20 +b22 +···+b2q−3) ifq ≡ 1(mod 4)
10 2
v = b(cid:98)q(1+b20 +b22 +···+b2q−3) ifq ≡ 3(mod 4)
I uv +u2v2 p(p−1)q(q−1) ?? u = (a20p +a22p +···+a2p−3p) ifp ≡ 1(mod 4)
11 2
u = (1+a20q +a22q +···+a2p−3q) ifp ≡ 3(mod 4)
I uv2 +u2v p(p−1)q(q−1) ?? v = (b20q +b22q +···+b2q−3q) ifq ≡ 1(mod 4)
12 2
v = (1+b20q +b22q +···+b2q−3q) ifq ≡ 3(mod 4)
8
F
4 Abelian codes of (C × C )
2 pm qn
Let p and q be distinct odd prime numbers such
that
gcd{p − 1, q − 1} = 2, gcd{p, q − 1} = 1,
gcd{p − 1, q} = 1 and
¯ Z Z
2 generates U( ) and U( ). (7)
2 2
p q
¯ Z
Notice that the condition 2 generates U( )
2
p
¯ Z
implies 2 generates U( ).
m
p
Theorem 4.1. Let p and q satisfy (7). Let
G = (cid:104)a(cid:105) × (cid:104)b(cid:105) be an abelian group of type
C × C , where C = (cid:104)a(cid:105) and C = (cid:104)b(cid:105).
m n m n
p q p q
F
Then the minimal codes of (C ×C ) are
m n
2 p q
described in the following table.
Ideal Primitive Dimension Code Remarks
Idempotent Weight
I a(cid:98)b 1 pmqn
0 (cid:98)
I a(b(cid:99)qj +b(cid:100)qj−1) qj−1(q −1) 2pmqn−j j = 1,...,n
0j (cid:98)
I (a(cid:99)pi +a(cid:100)pi−1)(cid:98)b pi−1(p−1) 2pm−iqn i = 1,...,m
i0
I∗ uv +u2v2 pi−1(p−1)qj−1(q−1) ?? u = a(cid:99)pi(a20pi−1 +a22pi−1 +···+a2p−3pi−1),
ij 2
if p ≡ 1(mod 4) or
u = a(cid:99)pi(1+a20pi−1 +a22pi−1 +···+a2p−3pi−1),
if p ≡ 3(mod 4) and
I∗∗ uv2 +u2v pi−1(p−1)qj−1(q−1) ?? v = b(cid:99)qj(b20qj−1 +b22qj−1 +···+b2q−3qj−1),
ij 2
if q ≡ 1(mod 4) or
v = b(cid:99)qj(1+b20qj−1 +b22qj−1 +···+b2q−3qj−1),
if q ≡ 3(mod 4)
9
Theorem 4.2. Let p , p and p be three
1 2 3
distinct positive odd prime numbers such that
gcd(p − 1, p − 1) = 2, for 1 ≤ i (cid:54)= j ≤ 3,
i j
¯ Z
and 2 generates the groups of units U( ).
p
i
Then a set of primitive idempotents of the
F
group algebra G for the finite abelian group
2
G = C × C × C , with C =< a >,
p p p p
1 2 3 1
C =< b > and C =< c >, is
p p
2 2
ˆ
e = aˆbcˆ
0
ˆ
e = aˆb(1 − cˆ)
1
ˆ
e = aˆ(1 − b)cˆ
2
ˆ
e = (1 − aˆ)bcˆ
3
2 2
e = (uv + u v )cˆ
4
2 2
e = (u v + uv )cˆ
5
2 2 ˆ
e = (uw + u w )b
6
2 2 ˆ
e = (u w + uw )b
7
2 2
e = (vw + v w )aˆ
8
2 2
e = (v w + vw )aˆ
9
ˆ 2 2 2
e = (1 − aˆ)(1 − b)(1 − cˆ) + u v w + uvw
10
ˆ 2 2 2
e = (1 − aˆ)(1 − b)(1 − cˆ) + u v w + uvw
11
ˆ 2 2 2 2
e = (1 − aˆ)(1 − b)(1 − cˆ) + u vw + uv w
12
and
ˆ 2 2 2
e = (1 − aˆ)(1 − b)(1 − cˆ) + uv w + u vw ,
13
10