Negacyclic codes of odd length over the ring $\mathbb{F}_p[u,v]/\langle u^2,v^2,uv-vu\rangle$

0.15 MB·English

by  Bappaditya Ghosh

NEGACYCLIC CODES OF ODD LENGTH OVER THE RING
F [u,v]/hu2,v2,uv−vui
p
BAPPADITYA GHOSH
Abstract. Wediscussthestructureofnegacycliccodesofoddlengthover
the ring F [u,v]/hu2,v2,uv−vui. We find the unique generating set, the
p
5 rank and the minimum distance for these negacyclic codes.
1
0
2

n
a 1. Introduction
J

9 The theory of error-correcting codes generally study the codes over the fi-
2 nite field. In recent time, the codes over the finite rings have been studied

extensively because of their important role in algebraic coding theory. Nega-
]
T
cyclic codes, an important class of constacyclic codes, over finite rings also
I
. have been well studied these day.
s
c In 1960’s, Berlekamp [1, 2] introduced negacyclic codes over the field F , p
p
[
odd prime, and designed a decoding algorithm that corrects up to t < p−1
2
1
Lee errors. Wolfmann [12], in 1999, studied negacyclic codes of odd(cid:0) length(cid:1)
v
Z
1 over 4. In 2003, Blackford [3] extended these study to negacyclic codes of
3 even length over Z .
4
4
The structure of negacyclic codes of length n over a finite chain ring such
7
0 that the length is not divisible by the characteristic of the residue field is
.
1 obtained by Dinh and Lo´pez-Permouth [6] in a more general setting in the
0 year 2004. When the length n of the code is divisible by the characteristic of
5
the residue field then the code is called a repeated-root codes. Repeated-root
1
: negacyclic codes over finite rings have also been investigated by many authors.
v
Xi The structure of negacyclic codes of length 2t over Z2m was obtained in [6]. In
2005, Dinh [4] investigated negacyclic codes of length 2s over the Galois ring
r
a GR(2a,m). S˘ala˘gean [11], in 2006, has studied the repeated-root negacyclic
codes over a finite chain ring and has shown that these codes are principally
generated over the Galois ring GR(2a,m). Various kinds of distances of nega-
cyclic codes of length 2s over Z are determined in [5]. The structure of the
2a
negacyclic codes of length 2ps over the ring F +uF have been discussed
pm pm
in [7].
Let p be a odd prime. In this paper we study the structure of negacyclic
codes of odd length over the non chain ring F [u,v]/hu2,v2,uv−vui. We find
p
a unique set of generators, rank and a minimal spanning set for these codes.
We also find the Hamming distance of these codes for length pl.
Email: [email protected]
Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, India.
1991 Mathematics Subject Classification. 94B15.
Key words and phrases. Negacyclic codes, Hamming distance.
1
2 B. Ghosh
The structures of cyclic codes over the ring Ru2,v2,p = Fp[u,v]/hu2,v2,uv−
vui have been discussed in [9]. We can view the cyclic and negacyclic codes
over the ring Ru2,v2,p asanideal inthe ringsRu2,v2,p[x]/hxn−1i and Ru2,v2,p[x]/
hxn+1irespectively. We define the ring isomorphism fromthe ring Ru2,v2,p[x]/
hxn−1i to the ring Ru2,v2,p[x]/hxn+1i to get the structure of negacyclic code
over the ring Ru2,v2,p.
2. Preliminaries
AlinearcodeC oflengthnoveraringRisnegacyclicif(−c ,c ,··· ,c )
n−1 0 n−2
∈ C whenever (c ,c ,··· ,c ) ∈ C. We can consider a negacyclic code
0 1 n−1
C of length n over a ring R as an ideal in the ring R[x]/hxn + 1i via the
correspondence Rn → R[x]/hxn + 1i, (c ,c ,··· ,c ) → c + c x + ··· +
0 1 n−1 0 1
cn−1xn−1. Let Ru2,v2,p = Fp +uFp +vFp +uvFp,u2 = 0, v2 = 0 and uv = vu.
This ring is isomorphic to the ring Fp[u,v]/hu2,v2,uv−vui. The ring Ru2,v2,p
is a finite commutative local ring with the unique maximal ideal hu,vi. The
set {{0},hui,hvi,huvi,hu+ αvi, hu,vi,h1i} gives list of all ideals of the ring
Ru2,v2,p, where α is a non zero element of Fp. Since the maximal ideal hu,vi
is not principal, the ring Ru2,v2,p is not a chain ring. The residue field R of
a ring R is define as R = R/M, where M is a maximal ideal. For the ring
Ru2,v2,p the residue field is Fp. Let µ : R[x] → R[x] denote the natural ring
homomorphism that maps r 7→ r+M and the variable x to x. We define the
degree of the polynomial f(x) ∈ R[x] as the degree of the polynomial µ(f(x))
in R[x], i.e., deg(f(x)) = deg(µ(f(x)) (see, for example, [10]). A polynomial
f(x) ∈ R[x] iscalled regular if itis not azero divisor. Thefollowing conditions
are equivalent for a finite commutative local ring R.
Proposition 2.1. (cf. [10, Exercise XIII.2(c)]) Let R be a finite commutative
local ring. Let f(x) = a +a x+···+a xn be in R[x], then the following are
0 1 n
equivalent.
(1) f(x) is regular;
(2) ha ,a ,··· ,a i = R;
0 1 n
(3) a is an unit for some i, 0 ≤ i ≤ n;
i
(4) µ(f(x)) 6= 0;
Let g(x) be a non zero polynomial in F [x]. By above proposition, it is easy
p
to see that the polynomial g(x) + up1(x) + vp2(x) + uvp3(x) ∈ Ru2,v2,p[x] is
regular. Note that deg(g(x)+up (x)+vp (x)+uvp (x)) = deg(g(x)).
1 2 3
3. Structures for negacyclic codes over the ring Ru2,v2,p
In this section we assume that n is an odd integer. Let Ru2,v2,p = Fp+uFp+
vF +uvF ,u2 = 0, v2 = 0 and uv = vu. The following theorem gives the ring
p p
isomorphism from the ring Ru2,v2,p[x]/hxn−1i to the ring Ru2,v2,p[x]/hxn +1i
Proposition 3.1. Let φ : Ru2,v2,p[x] → Ru2,v2,p[x] be a map defined as φ(f(x)) =
hxn−1i hxn+1i
f(−x), for all f(x) ∈ Ru2,v2,p[x]. The map φ is a ring isomorphism.
hxn−1i
Negacyclic codes of odd length overthering Ru2,v2,p 3
Proof. For polynomials f(x),g(x) ∈ Ru2,v2,p[x],
f(x) ≡ g(x) mod (xn −1);
if and only if there exists a polynomial h(x) ∈ Ru2,v2,p[x] such that
f(x)−g(x) = h(x)(xn −1);
if and only if
f(−x)−g(−x) = h(−x)((−x)n −1) = −h(−x)(xn +1);
if and only if
f(−x) ≡ g(−x) mod (xn +1);
This implies that for f(x),g(x) ∈ Ru2,v2,p[x], φ(f(x)) = φ(g(x)) if and only if
hxn−1i
f(x) = g(x). Hence, φ is well-defined and one-to-one. Since the rings Ru2,v2,p[x]
hxn−1i
and Ru2,v2,p[x] are finite and of same order, φ is an onto map. It is easy to see
hxn+1i
that φ is a ring homomorphism. So φ is a ring isomorphism. (cid:3)
Remark 3.2. We restrict the isomorphism φ to an isomorphism φ : Fp[x] →
hxn−1i
Fp[x] .
hxn+1i
Throughoutthispaperweusetheisomorphismφandrestrictionofφdefined
in Proposition 3.1 and Remark 3.2.
Proposition 3.3. Let R be a ring. Let A ⊆ R[x] , B ⊆ R[x] be two sets
hxn−1i hxn+1i
such that φ(A) = B. Then A is an ideal of R[x] if and only if B is an ideal
hxn−1i
of R[x] . Equivalently, A is a cyclic code of length n over the ring R if and
hxn+1i
only if B is a negacyclic code of length n over the ring R.
Proof. The proof is obvious since the map φ is a ring isomorphism. (cid:3)
Theorem 3.4. Let C be a negacyclic code of length n over the ring Ru2,v2,p.
Then C will be of the form C = hg (x)+ug (x)+vg (x)+vug (x),ug (x)+
1 11 12 13 2
vg (x)+vug (x),vg (x)+vug (x),vug (x)i, where g (x)|g (x)|g (x)|(xn+1)
22 23 3 33 4 4 2 1
and g (x)|g (x)|g (x)|(xn +1).
4 3 1
Proof. The code C is a negacyclic codes of length n over the ring Ru2,v2,p.
From Proposition 3.3, we know that for the negacyclic code C there exist
a cyclic code, say A over the same ring and of same length. We know the
structure of a cyclic code over the ring Ru2,v2,p from [9]. Let the cyclic code
over the ring Ru2,v2,p be A = hg(x) + up1(x) + vq1(x) + vur1(x),ua1(x) +
vq (x)+vur (x),va (x)+vur (x),vua (x)i, where a (x)|a (x)|g(x)|(xn −1)
2 2 2 3 3 3 1
and a (x)|a (x)|g(x)|(xn−1). Now the polynomials g(x)+up (x)+vq (x)+
3 2 1 1
vur (x), ua (x)+vq (x)+vur (x), va (x)+vur (x), vua (x) ∈ R[x] . There-
1 1 2 2 2 3 3 hxn−1i
fore from the definition of φ from Proposition 3.1, we get φ(g(x)+up (x) +
1
vq (x)+vur (x)) = g(−x)+up (−x)+vq (−x)+vur (−x) = g (x)+ug (x)+
1 1 1 1 1 1 11
vg (x) + vug (x), φ(ua (x) + vq (x) + vur (x)) = ua (−x) + vq (−x) +
12 13 1 2 2 1 2
vur (−x) = ug (x) + vg (x) + vug (x), φ(va (x) + vur (x)) = va (−x) +
2 2 22 23 2 3 2
vur (−x) = vg (x) + vug (x), φ(vua (x)) = vua (−x) = vug (x) and
3 3 33 3 3 4
4 B. Ghosh
φ(xn−1) = −(xn+1), where g(−x) = g (x), a (−x) = g (x), a (−x) = g (x),
1 1 2 2 3
a (−x) = g (x), p (−x) = g (x), q (−x) = g (x), r (−x) = g (x), q (−x) =
3 4 1 11 1 12 1 13 2
g (x), r (−x) = g (x), r (−x) = g (x). Again φ(A) = C. Therefore the
22 2 23 3 33
code C can be written as C = hg (x)+ug (x)+vg (x)+vug (x),ug (x)+
1 11 12 13 2
vg (x)+vug (x),vg (x)+vug (x),vug (x)i, where g (x)|g (x)|g (x)|(xn+1)
22 23 3 33 4 4 2 1
and g (x)|g (x)|g (x)|(xn +1). (cid:3)
4 3 1
Theorem 3.5. Any negacyclic code C of length n over the ring Ru2,v2,p is
uniquely generated by the polynomials A = g (x)+ug (x)+vg (x)+uvg (x),
1 1 11 12 13
A = ug (x) + vg (x) + uvg (x),A = vg (x) + uvg (x),A = uvg (x),
2 2 22 23 3 3 33 4 4
where, g (x) are zero polynomial or deg(g (x)) < deg(g (x)) for 1 ≤ i ≤ 3,
ij ij j+1
i ≤ j ≤ 3.
Proof. The code C is generated by the polynomial A ,A ,A and A . Let
1 2 3 4
A = hg(x) + up (x) + vq (x) + vur (x),ua (x) + vq (x) + vur (x),va (x) +
1 1 1 1 2 2 2
vur3(x),vua3(x)i be the cyclic code over the ring Ru2,v2,p such that φ(A) = C.
Now, for any polynomial f(x) ∈ F [x] we get deg(f(−x)) = deg(f(x)). There-
p
fore, deg(φ(f(x))) = deg(f(x)). From above theorem, we have φ(g(x)) =
g (x), φ(a (x)) = g (x), φ(a (x)) = g (x), φ(a (x)) = g (x), φ(p (x)) =
1 1 2 2 3 3 4 1
g (x), φ(q (x)) = g (x), φ(r (x)) = g (x), φ(q (x)) = g (x), φ(r (x)) =
11 1 12 1 13 2 22 2
g (x), φ(r (x)) = g (x). Also from Theorem 3.1 of [9], we have deg(p (x)) <
23 3 33 1
deg(a (x)), deg(q (x)) < deg(a (x)), deg(r (x)) < deg(a (x)), deg(q (x)) <
1 1 2 1 3 2
deg(a (x)), deg(r (x)) < deg(a (x)), deg(r (x)) < deg(a (x)). Therefore,
2 2 3 3 3
from the relation deg(φ(f(x))) = textdeg(f(x)) we can write deg(φ(p (x))) <
1
deg(φ(a (x))), deg(φ(q (x))) < deg(φ(a (x))), deg(φ(r (x))) < deg(φ(a (x))),
1 1 2 1 3
deg(φ(q (x))) < deg(φ(a (x))),deg(φ(r (x))) < deg(φ(a (x))),deg(φ(r (x))) <
2 2 2 3 3
deg(φ(a (x))). This implies that deg(g (x)) < deg(g (x)) for 1 ≤ i ≤ 3,
3 ij j+1
i ≤ j ≤ 3. To prove uniqueness we assume that the polynomial A′ =
1
g (x)+ug′ (x)+vg′ (x)+uvg′ (x) ∈ C satisfies degree result deg(g′ (x)) <
1 11 12 13 1j
deg(g (x)) for 1 ≤ j ≤ 3. Since, φ(A) = C, therefore, there exists a poly-
j+1
nomial g(x) + up′(x) + vq′(x) + vur′(x) ∈ A, such that φ(g(x) + up′(x) +
1 1 1 1
vq′(x) + vur′(x)) = g (x) + ug′ (x) + vg′ (x) + uvg′ (x), where φ(p′(x)) =
1 1 1 11 12 13 1
p′(−x) = g′ (x), φ(q′(x)) = q′(−x) = g′ (x), φ(r (x)) = r′(−x) = g′ (x).
1 11 1 1 12 1 13
Since, deg(φ(f(x))) = deg(f(x)), for all f(x) ∈ A, thus, deg(φ(p′(x))) =
1
deg(p′(x)) = deg(g′ (x)). Similarly, deg(q′(x)) = deg(g′ (x)), deg(r′(x)) =
1 11 1 12 1
deg(g′ (x)). Therefore, deg(p′(x)) = deg(g′ (x)) < deg(g (x)) = deg(a (x)).
13 1 11 2 1
This implies that deg(p′(x)) < deg(a (x)) Similarly we get, deg(q′(x)) <
1 1 1
deg(a (x)), deg(r′(x)) < deg(a (x)). But from Theorem 3.1 of [9], we know
2 1 3
that the polynomial g(x) + up (x) + vq (x) + vur (x) ∈ A is unique which
1 1 1
satisfying the degree result. Therefore, p (x) = p′(x), q (x) = q′(x) and
1 1 1 1
r (x) = r′(x). This implies that φ(p (x)) = φ(p′(x)), thus g (x) = g′ (x).
1 1 1 1 11 11
Similarly, g (x) = g′ (x) and g (x) = g′ (x). Hence, A is unique. Similarly
12 12 13 13 1
we can prove that A ,A and A are also unique.
2 3 4
(cid:3)
Theorem 3.6. Let C = hg (x)+ug (x)+vg (x)+vug (x),ug (x)+vg (x)+
1 11 12 13 2 22
vug (x),vg (x)+vug (x),vug (x)i be a negacyclic code of length n over the
23 3 33 4
Negacyclic codes of odd length overthering Ru2,v2,p 5
ring Ru2,v2,p. Then we must have the following properties
g (x)|g (x)|g (x),g (x)|g (x)|g (x)|(xn +1), (1)
4 3 1 4 2 1
xn +1
g (x)| g (x), for 1 ≤ i ≤ 3, (2)
i+1 ii
g (x)
i
g (x)
1
g (x)| g (x) (3)
3 22
g (x)
2
g (x)|g (x) (4)
4 22
g (x)
1
g (x)| g (x)− g (x) (5)
4 11 33
(cid:18) g (x) (cid:19)
3
g (x) g (x)
1 1
g (x)| g (x)− g (x)+ g (x)g (x) (6)
4 12 23 22 33
(cid:18) g (x) g (x)g (x) (cid:19)
2 2 3
xn +1
g (x)| s for 1 ≤ i ≤ 2 and for a fix i for 1 ≤ j ≤ 3−i,
i+j+1 i(i+j)
g (x)
i
j
s
i(i+l−1)
where, s = g and s = g − g (x). (7)
ii ii i(i+j) i(i+j) (i+l)(i+j)
g (x)
Xl=1 i+l
Proof. Let A = hg(x)+up (x)+vq (x)+vur (x),ua (x)+vq (x)+vur (x),
1 1 1 1 2 2
va2(x)+vur3(x),vua3(x)i be the cyclic code over the ring Ru2,v2,p such that
φ(A) = C. Also, from Theorem 3.4, we have φ(g(x)) = g (x), φ(a (x)) =
1 1
g (x), φ(a (x)) = g (x), φ(a (x)) = g (x), φ(p (x)) = g (x), φ(q (x)) =
2 2 3 3 4 1 11 1
g (x), φ(r (x)) = g (x), φ(q (x)) = g (x), φ(r (x)) = g (x), φ(r (x)) =
12 1 13 2 22 2 23 3
g (x). Now, from Remark 3.2, we get that the map φ, φ : Fp[x] → Fp[x]
33 hxn−1i hxn+1i
such that φ(f(x)) = f(−x), ∀ f(x) ∈ Fp[x] is an isomorphism. Now from
hxn−1i
Proposition 3.2 of [9], we know that the properties are true for the ring Fp[x] .
hxn−1i
Therefore all of these properties are true for the ring Fp[x] . (cid:3)
hxn+1i
The following theorem characterizes the free negacyclic codes over the ring
Ru2,v2,p.
Theorem 3.7. If C = hg (x)+ug (x)+vg (x)+vug (x),ug (x)+vg (x)+
1 11 12 13 2 22
vug (x),vg (x)+vug (x),vug (x)i be a negacyclic code of length n over the
23 3 33 4
ring Ru2,v2,p, then C is a free negacyclic code if and only if g1(x) = g4(x). In
this case, we have C = hg (x) + ug (x) + vg (x) + vug (x)i and g (x) +
1 11 12 13 1
ug11(x)+vg12(x)+vug13(x)|(xn +1) in Ru2,v2,p.
Proof. We are given that C is a negacyclic code over the ring Ru2,v2,p. Hence
from Proposition 3.3, there exist one and only one cyclic code A over the ring
Ru2,v2,p such that φ(A) = C. Let the cyclic code be A = hg(x) + up1(x) +
vq (x)+vur (x),ua (x)+vq (x)+vur (x),va (x)+vur (x),vua (x)i, where
1 1 1 2 2 2 3 3
a (x)|a (x)|g(x)|(xn − 1) and a (x)|a (x)|g(x)|(xn − 1). Therefore we have,
3 1 3 2
φ(g(x)+ up (x) +vq (x) +vur (x)) = g (x) +ug (x) + vg (x) + vug (x),
1 1 1 1 11 12 13
φ(ua (x) + vq (x) + vur (x)) = ug (x) + vg (x) + vug (x), φ(va (x) +
1 2 2 2 22 23 2
vur (x)) = vg (x)+vug (x), φ(vua (x)) = vug (x)andφ(xn−1) = −(xn+1).
3 3 33 3 4
Now, it is given g (x) = g (x). Since φ is an isomorphism therefore g(x) =
1 4
6 B. Ghosh
a (x). We know from Proposition 3.3 of [9] that A = hg(x) + up (x) +
3 1
vq (x)+vur (x)i if and only if g(x) = a (x). Now φ(g(x)+up (x)+vq (x)+
1 1 3 1 1
vur (x)) = g (x)+ug (x)+vg (x)+vug (x), Hence C = hg (x)+ug (x)+
1 1 11 12 13 1 11
vg (x)+vug (x)i. Again we have φ(xn−1) = −(xn+1) and we know from
12 13
Proposition 3.3 of [9] that g(x)+up (x)+vq (x)+vur (x)|(xn −1). Hence
1 1 1
g (x)+ug (x)+vg (x)+vug (x)|(xn +1). (cid:3)
1 11 12 13
Note that we get the simpler form for the generators of the negacyclic code
over Ru2,v2,p, like in the above theorem, if we have g1(x) = g2(x),g3(x) or
g (x) = g (x),g (x).
4 2 3
In the following theorem we write the structure of C when n be relatively
prime to p.
Theorem 3.8. Let C be a negacyclic code over the ring Ru2,v2,p of length n. If
n is relatively prime to p, then we have C = hg (x)+ug (x)+uvg (x),vg (x)+
1 2 13 3
uvg (x)i with g (x)|g (x)|(xn +1) and g (x)|g (x)|g (x)|(xn +1).
4 2 1 4 3 1
Proof. Let C be a negacyclic code over the ring Ru2,v2,p. Hence from Propo-
sition 3.3, there exists one and only one cyclic code A over the ring Ru2,v2,p
such that φ(A) = C. If n is relatively prime to p, then from Theorem 3.4
of [9], we can write the cyclic code A = hg(x) + ua (x) + uvr (x),va (x) +
1 1 2
uva (x)i with a (x)|g(x)|(xn−1) and a (x)|a (x)|g(x)|(xn−1). Let φ(g(x)+
3 1 3 2
ua (x) + uvr (x)) = g (x) + ug (x) + uvg (x) and φ(va (x) + uva (x)) =
1 1 1 2 13 2 3
vg (x) + uvg (x), where, g(−x) = g (x), r (−x) = g (x), a (−x) = g (x),
3 4 1 1 13 1 2
a (−x) = g (x), a (−x) = g (x). Therefore C can be written as C =
2 3 3 4
hg (x) + ug (x) + uvg (x),vg (x) + uvg (x)i with g (x)|g (x)|(xn + 1) and
1 2 13 3 4 2 1
g (x)|g (x)|g (x)|(xn +1). (cid:3)
4 3 1
4. The Ranks and the minimum distance
In this section, we find the rank and minimal spanning set of negacyclic
codes over the ring Ru2,v2,p. Following Dougherty and Shiromoto [8, page
401], we define the rank of the code C by the minimum number of generators
of C and define the free rank of C by the maximum of the ranks of Ru2,v2,p-free
submodules of C.
Theorem 4.1. Let n be not relatively prime to p. Let C be a negacyclic
code over the ring Ru2,v2,p of length n. If C = hg1(x) + ug11(x) + vg12(x) +
vug (x),ug (x)+vg (x)+vug (x),vg (x)+vug (x),vug (x)i with deg(g (x))
13 2 22 23 3 33 4 1
= r , deg(g (x)) = r , deg(g (x)) = r , deg(g (x)) = r , then the minimal
1 2 2 3 3 4 4
spanning set of C is B = {A ,xA ,··· ,xn−r1−1A ,A ,xA ,··· ,xr1−r2−1A ,
1 1 1 2 2 2
A ,xA ,··· ,xr1−r3−1A ,A ,xA ,··· ,xr′−r4−1A }, where, r′ = min{r ,r } and
3 3 3 4 4 4 2 3
A = g (x)+ug (x)+vg (x)+vug (x), A = ug (x)+vg (x)+vug (x),
1 1 11 12 13 2 2 22 23
A = vg (x) +vug (x), A = vug (x) also C has free rank n−r and rank
3 3 33 4 4 1
n+r +r′ −r −r −r .
1 2 3 4
Proof. Let C be a negacyclic code over the ring Ru2,v2,p of length n, where
n is not relatively prime to p. From the Proposition 3.3, we get that there
Negacyclic codes of odd length overthering Ru2,v2,p 7
exists a cyclic code A over the same ring such that φ(A) = C. Now, we
know from Theorem 4.1 of [9], the minimal spanning set of the cyclic code A
over the ring Ru2,v2,p is {g(x)+up1(x)+vq1(x)+vur1(x),x(g(x)+up1(x)+
vq (x) + vur (x)),··· ,xn−r1−1(g(x) + up (x) + vq (x) + vur (x)),ua (x) +
1 1 1 1 1 1
vq (x)+vur (x),x(ua (x)+vq (x)+vur (x)),··· ,xr1−r2−1(ua (x)+vq (x)+
2 2 1 2 2 1 2
vur (x)),va (x)+vur (x),x(va (x)+vur (x)),··· ,xr1−r3−1(va (x)+vur (x)),
2 2 3 2 3 2 3
vua (x),x(vua (x)),··· ,xr′−r4−1(vua (x))}, where, r′ = min{r ,r }. From
3 3 3 2 3
Theorem 3.4 we have φ(g(x)+ up (x) + vq (x) + vur (x)) = A ,φ(ua (x) +
1 1 1 1 1
vq (x) + vur (x)) = A ,φ(va (x) + vur (x)) = A and φ(vua (x)) = A .
2 2 2 2 3 3 3 4
Therefore the spanning set of negacyclic code C over the ring Ru2,v2,p is B =
{A ,xA ,··· ,xn−r1−1A ,A ,xA ,··· ,xr1−r2−1A ,A ,xA ,··· ,xr1−r3−1A ,A ,
1 1 1 2 2 2 3 3 3 4
xA ,··· ,xr′−r4−1A }, where, r′ = min{r ,r } and A = g (x) + ug (x) +
4 4 2 3 1 1 11
vg (x)+vug (x), A = ug (x)+vg (x)+vug (x), A = vg (x)+vug (x),
12 13 2 2 22 23 3 3 33
A = vug (x). (cid:3)
4 4
Let n be a positive integer not relatively prime to p. Let C be a negacyclic
code of length n over the ring Ru2,v2,p. We know that there exists a cyclic
code A of length n over the ring Ru2,v2,p such that φ(A) = C, where, φ is
defined as φ(f(x)) = f(−x), for f(x) ∈ A. The following lemma shows that
the isomorphism φ is a distance preserving map.
Lemma 4.2. Let φ(A) = C, where, A and C are the cyclic and negacyclic
codes of length n over the ring Ru2,v2,p and φ is defined as φ(f(x)) = f(−x),
for f(x) ∈ A, then, w (f(x)) = w (φ(f(x))).
H H
Proof. Let f(x) = in=−01cixi, where, ci ∈ Ru2,v2,p. Now φ(f(x)) = f(−x) =
n−1(−1)ic xi. ThPerefore the coefficient of xi of f(x) and f(−x) are c and
i=0 i i
P(−1)ici. That is both coefficient are simultaneously 0 or non 0. Hence,
w (f(x)) = w (φ(f(x))). (cid:3)
H H
Theorem 4.3. Let n be not relatively prime to p. If C = hA ,A ,A ,A i is
1 2 3 4
a negacyclic code of length n over the ring Ru2,v2,p. Then wH(C) = wH(A),
where, A is the cyclic codes over the ring Ru2,v2,p such that φ(A) = C.
Proof. Let h(x) be the minimum weighted polynomial in A and the weight is
w (h(x)) = m. There exists apolynomial f(x) ∈ C such thatφ(h(x)) = f(x).
H
From Lemma 4.2, the weight of w (f(x)) = m. Now we prove that f(x) is the
H
minimum weighted polynomial in C. If possible, let f (x) be the minimum
1
weighted polynomial of C and w (f (x)) < m. There exists a polynomial
H 1
h (x) ∈ A such that φ(h (x)) = f (x). Again, from Lemma 4.2, w (h (x)) =
1 1 1 H 1
w (f (x)) < m. Hence, a contradiction that h(x) be the minimum weighted
H 1
polynomial in A. Therefore, w (C) = w (A). (cid:3)
H H
Definition 4.4. Let m = b pl−1+b pl−2+···+b p+b , b ∈ F ,0 ≤ i ≤
l−1 l−2 1 0 i p
l −1, be the p-adic expansion of m.
(1) If b 6= 0 for all 1 ≤ i ≤ q,q < l, and b = 0 for all i,q + 1 ≤ i ≤ l,
l−i l−i
then m is said to have a p-adic length q zero expansion.
8 B. Ghosh
(2) If b 6= 0 for all 1 ≤ i ≤ q,q < l, b = 0 and b 6= 0 for some
l−i l−q−1 l−i
i,q+2 ≤ i ≤ l, then m is said to have p-adic length q non-zero expansion.
(3) If b 6= 0 for 1 ≤ i ≤ l, then m is said to have a p-adic length l expansion
l−i
or p-adic full expansion.
The following theorem follows from the above theorem and Theorem 5.4 of
[9].
Theorem 4.5. Let C be a negacyclic code over the ring Ru2,v2,p of length pl
where l is a positive integer. Then, C = hA ,A ,A ,A i, where, g (x) =
1 2 3 4 1
(x + 1)t1,g (x) = (x + 1)t2,g (x) = (x + 1)t3,g (x) = (x + 1)t4, for some
2 3 4
t > t > t > 0, t > t > t > 0 (whereA ’s and g ’s are defined in Theorem
1 2 4 1 3 4 i i
3.5)
(1) If t ≤ pl−1, then d(C) = 2.
4
(2) If t > pl−1, let t = b pl−1+b pl−2+···+b p+b be the p-adicexpansion
4 4 l−1 l−2 1 0
of t and g (x) = (x+1)t4 = (xpl−1+1)bl−1(xpl−2+1)bl−2···(xp1+1)b1(xp0+
4 4
1)b0.
(a) If t has a p-adic length q zero expansion or full expansion (l = q),
4
then d(C) = (b +1)(b +1)···(b +1).
l−1 l−2 l−q
(b) If t has a p-adic length q non-zero expansion, then d(C) = 2(b +
4 l−1
1)(b +1)···(b +1).
l−2 l−q
5. Examples
Example 5.1. Negacyclic codes of length 5 over Ru2,v2,5 = F5 +uF5 +vF5 +
uvF ,u2 = 0,v2 = 0,uv = vu: We have
5
x5 +1 = (x+1)5 over Ru2,v2,5.
Let g = x + 1 and c ,c ,c ,c ,c ,c ∈ F . The non-zero negacyclic codes of
0 1 2 3 4 5 5
length 5 over Ru2,v2,5 with generator polynomial, rank and minimum distance
are given in tables below.
Table 1. All non zero free negacyclic codes of length 5 over Ru2,v2,5.
Non-zero generator polynomials Rank d(C)
< g4 +uc g3 +vc g3 +uvc g3 >, c c = 0 1 5
0 1 2 0 1
< g3 +uc g2 +vc g2 +uv(c +c x)g > 2 4
0 1 2 3
< g2 +u(c +c x)+v(c +c x)+uv(c +c x) >, 3 3
0 1 2 3 4 5
c = c or c = c
0 1 2 3
< g +uc +vc +uvc > 4 2
0 1 2
< 1 > 5 1
Negacyclic codes of odd length overthering Ru2,v2,p 9
Table 2. All non zero non free single generated negacyclic codes of length
5 over Ru2,v2,5.
Non-zero generator polynomials Rank d(C)
< ug4+vc g4+uvc g3 > 1 5
0 1
< vg4 +uvc g3 > 1 5
0
< uvg4 > 1 5
< ug3+v(c +c x)g3 +uv(c +c x)g > 2 4
0 1 2 3
< vg3 +uv(c +c x)g > 2 4
0 1
< uvg3 > 2 4
< ug2+v(c +c x+c x2)g2 +uv(c +c x) > 3 3
0 1 2 3 4
< vg2 +uv(c +c x) > 3 3
0 1
< uvg2 > 3 3
< ug +v(c +c x+c x2 +c x3)g +uvc > 4 2
0 1 2 3 4
< vg +uvc > 4 2
0
< uvg > 4 2
< u+v(c +c x+c x2 +c x3 +c x4) > 5 1
0 1 2 3 4
< v > 5 1
< uv > 5 1
Table 3. Some non zero non free negacyclic codes of length 5 over Ru2,v2,5.
Non-zero generator polynomials Rank d(C)
< g4 +uc g3 +vc g3 +uvc g2,uvg3 > 2 4
0 1 2
< ug4 +uvc g3,vg4+uvc g3 > 2 5
0 1
< ug4 +v(c +c x)g3 +uvg2,uvg3 > 2 4
0 1
< ug4 +vc g3 +uvc g2,vg4 > 2 5
0 1
< ug4 +uvc g2,uvg3 > 2 4
0
< vg4 +uvc g2,uvg3 > 2 4
0
< vg4 +uvc g,uvg2 > 3 3
0
< g3 +uc g +vc g +uvc ,ug2+vc g +uvc , 5 2
0 1 2 3 4
vg2+uvc ,uvg >, c c = 0
5 0 2
< ug3 +v(c +c x)g3 +uv(c +c x),uvg2 > 3 3
0 1 2 3
< vg3 +uvc ,uvg > 4 2
0
< g2 +uc +vc ,ug+vc ,vg,uv > 6 1
0 1 2
< ug2 +vc +uvc ,vg2+uvc ,uvg > 7 2
0 1 2
< vg2 +uvc ,uvg > 4 2
2
< g +uc +vc ,uv > 5 1
0 1
< g +uc ,v > 5 1
0
< g +vc ,u+vc > 5 1
0 1
< g,u,v > 6 1
< ug +vc ,vg,uv > 9 1
0
< vg,uv > 5 1
< u,v > 10 1
10 B. Ghosh
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