Open Journal of Applied Sciences, 2013, 3, 27-31
doi:10.4236/ojapps.2013.31B1006 Published Online April 2013 (http://www.scirp.org/journal/ojapps)
Images of Linear Block Codes over qqq q
F
uF vFuvF

Jane D. Palacio, Virgilio P. Sison
Institute of Mathematical Sciences and Physics, University of the Philippines Los Baños, College, Laguna, Philippines
Email: jdpalacio@uplb.edu.ph, vpsison@uplb.edu.ph
Received 2013
ABSTRACT
In this paper, we considered linear block codes over 22
,0,
qqq qq
RFuFvFuvFuv uvvu
  where ,
m
qp
m
. First we looked at the structure of the ring. It was shown that is neither a finite chain ring nor a principal
ideal ring but is a local ring. We then established a generator matrix for the linear block codes and equipped it with a
homogeneous weight function. Field codes were then constructed as images of these codes by using a basis of over
q
R
q
R
q
F
. Bounds on the minimum Hamming distance of the image codes were then derived. A code meeting such bounds is
given as an example.
Keywords: -ary Images; Distance Bounds
q
1. Introduction
Let be a prime number, p,m
and q
m
qp
F
de-
note the Galois field with elements. During the late
1990s, C. Bachoc used linear block codes over ,
q
p
p
F
uF
2
u0 for constructing modular lattices. Its success
motivated the study of linear block codes over the finite
chain ring
p
p
F
uF. And many of the results from these
studies have been extended over finite chain rings of the
form
21
..., 0,
rr
qq qq
FuFuFuFu r
 
.
Such rings can be seen as natural extensions of qq
F
uF
.
Another ring extension of qq
F
uF
is
qqq q
RFuFvFuvFq
where Unlike
22
0, .uv uvvu,
qq
F
uF is neither
q
R
a finite chain ring nor a principal ideal ring. B. Yildiz and
S. Karadeniz introduced linear block codes over the ring
222 2
F
uF vF uvF in [6]. Self-dual codes, cyclic codes
and constacyclic codes over this ring were also studied
by these authors in [3,7,8]. In 2011, X. Xu and X. Liu
studied the structure of cyclic codes over in
[5].
q
R
In this work, we will analyze linear block codes over
q
R. The structure of the ring will be discussed in Section
2. The generator matrix of linear block codes over
and weight functions defined on will be tackled in
q
R
q
R
Section 3. The -ary images of these linear block codes
and bounds on its minimum Hamming distance will be
presented in Sections 4 and 5, respectively. Lastly, a
code meeting these bounds is given in Section 6.
2. Preliminaries and Definitions
q
Structure of the Ring qqq
F
uF vF q
uvF
Let q
R denote the ring qqqq
F
uF vF uvF
abu
w
ents can be uniquely written as
hose
elem cvduv
where ,,,q
abcd F
. An element of q
R is a unit if and
only if 0a
. The ring has 5q ideals namely
0,
, ,,
q
uvvR u jv wq
jF. q
R is not ,,u vhere a
principal ideal ring since the maxiemal idal
,uv is
generated by u and v. The cardinality of the iare deals
,uv q
2,vujq

3
,,uv q and 4.
q
Rq v
Its lattice of ideals is shown in Figur 1. As can be
in the lattice of ideals,is not a finite chain ring. But
eseen
q
R
it is a local, Noetherian and Artinian ring. All zero divi-
sors are the elements of

,\0uv and its units are the
elements of
\,
q
Ruv.
Figure 1. Lattice of ideals of q
qqq
FuFvFuvF .
Copyright © 2013 SciRes. OJAppS
J. D. PALACIO, V. P. SISON
28
Clearly, the ring is isomorphic to

22
[, ],,
q
F
x yxyxyyx
ring of all 44 ma
. It is also isomorphic to the
trices of the form
Moreover, is
Frobenius with generating character
00
.
00
000
abcd
ac
ab
a






q
R

2
:,
itrd
p
qTabucvduv
trace map on
R e
 wher r denotes
the
e t
q
F
and T is the multiplicative group of
unlex num
Further, is a vector
it compbers.
q
R space over q
F
with dimension
4. A basis over q
of Rq
F
is given by the
is
set
which the polynomial
Another sideren this work
3. lo
Any linear block code over a fi
has a generator matrix which can be pu
l
(1
where

1,, ,uvuv
basis of q
R.
we will refer to as
conbasis d i

1,1,1,1uv uvv uvu uvuv.
Linear Bck Codes over qqq q
FuFvFuvF
nite commutative ring R
t in the following
form
1
11,2k
aI A
2
1,3 1,1
1
,1
l ll
AA
22 2,32 2,

l
kl
lk
aI aA aA
G
aI aA




)
,ij
A
are binary matrices for and are trices
over for. A code of this form has
1ima
q
R 1i
1
i
lk
i
i
aR
elements, where the '
i
as define th e nonzero equiva-
lence classes
12
,,,
l
aa a under the equivalen
relation on R defined by
ce
if for a unit in ababuuR

for some
ii
aRxxarrR ; and the blanks in G are
zeros.
B of length n over is an
-submodule of. B has a generator mwhich
be put in shown in Figure 2e
to be filled with
A linear block code q
R
wher
q
R
can
n
q
R
the form
atrix
,ij
A
are ij
kk
matrover , ices q
R,ij
D are ij
kk
matri-
ces over 2
F
and the blank parts of
GB
4k
are to be
k
filled with zeros. Moreover, B has 3
1q
qqq
2t

dewords where
2
2
q
i
i
k
t
co
. A linear block code over
q
R i if and only if 0
i
k r all 3iq.
Now, w equip B with two weight funs namely
the usual Hamming metric and a homogeneous wei
function.
s freefo
e
ght
Honold
ring with generating character
2, ,
nctio
Lemma 2.1. (T. , [2]) Let R be a Frobenius
, then every homoge-
ne an be in terms of ous weight whom on R c expressed
as follows

hom 1
yR
wxy
R
1

(1)
where R
is the group of units of R.
Theorem 2.1. A homogeneous weight whom on Rq is
given by


 
hom 1
wx ifx
q

\
quv
\0
if xR
quv

(2)
Proof: Let
0otherwise
q
x
abucvduvR
 
ma, every homogeneous wei
. Now, using the
previous lemght on Rq can
be expressed as
 
hom 3
1
11yR
wx
qq
y

Case 1. Let q
x
R
. There are

2
1qq units having
the same d, for any q
dF
1m
p
. But there are ele-
ments of Fqr an that has trace j, foy
p
jF. He
rq
Figure 2. Generator Matrix of Linear Block Codes over q
nce,

4
GB

3
()uv
I u

2, 4q
uD
 
 
1
3
1, 21,31, 41, 51, 4
2,4
3,43,53,4
4,54, 4
,1,4
3,4
()
r
q
k q
k q
k q
krr
kqq
IA AAAA
vI vDvDvDvD
uD
vD uvD
ujvI ujvDujvD
uvI uvD


 
 
 
 
 
 
 
 
 
 
 

 


22,3 2,5k
uI uD
qqq
F
uF vF uvF
 .
Copyright © 2013 SciRes. OJAppS
J. D. PALACIO, V. P. SISON 29
 

2
21
1.
p
q
i
j
mp
jF
yR
xyqq pe


But
2
0
p
ij
p
jF
e
. So,
Case 2. Let. For every
hom
w.
 
\0xuvq
aF
cv duv
, there
are 3
q units of the form yabu

e trace value j, f
. Now,
of these have the samor any
1
m
p
p
jF while there are 11
m
p
of them with trace zero.
Hence,

 
2ij
31 31
1.
qq
mm
p
yR jF
xyqpeq p





But
2
1
q
ij
p
jF
e

. So, hom 1
q
wq

.
Case 3. Let

,\
x
uv uv. There are 1q
ele-
moe
ents of

,\uv ve the same cent for

uv that ha
,\
ffici
uv. For each element

x
uv appears copies
e multiset

uv q
in th
,,\
q
x
yyR xuvuv
 . Moreover,
there are 1m
p elements of q
F
that has trace j, for any
p
jF. Hence


2
1
1.
q
q
ij
mp
jF
yR
xyqq pe
 

xtend this to turally: if We en
q
R na
12
,,,
n
x
xx x
then . The homogeneous (resp.


hom hom
1
n
i
wx wx
distance be
i
Hamming)tween any distinct vectors ,,
n
q
x
yR
denoted by

hom ,dxy (resp.
y), is defined as

hom
wxy (resp.
weote the mini-
mum homogeneong) distance
,
H
dx
We will d
resp. Ham
H
us
xy)
distance
. n
( mi
by a linear block code over by (resp.
q
R hom
d
H
d).
4. The q-ary Images of Linear Block Codes
over qqqq
F
uF vFuvF
Let ents of an ordered basis
of y element ofcan be written in the
form
1
bb
q
R. Then an
4
234
,,,bb be distinct elem
q
R
1
,
ii iq
i
ab aF
. Define the mapping

4
12 4
1
:
,,,
qq
ii
i
RF
aba aaa
3
We then extend
tn
q
R dinate-w: o cooriseif
12
,,,
n
x
xx x and ,
1
ii
jj
4
j
x
ab
then
1,1 ,, , ,,,, ,,
1,42,12,4,1,4nn
x
aaa aa a.
It is easy to show that
isn q
a
F
-module isomor-
phism
The B is a linear block c
.
orem 4.1. Ifode over
of length n, then

q
R

Bxx

 Bis a linear block
code over q
F
with
Proof: First we show that for every
length 4n.
,q
4n
x
Bx
F.
12
,, n
x
xx x B
, Since

4
iq
x
F for any
Let
1, 2,,,ni
then
4.
n
q
x
F
Net we show that x
B
q
is a subspace of 4n
F
. Let q
s
F and let
1
,
y
yB
.
Then there exist 1
,
x
xB
such that
y
x
and
11
y
x
. But

11
s
yy s
xx since
is a mod-
ule homomorphism. Since 1
s
xx B, 
1
s
yy B
 .
Thus,
B
is a subspacef 4n
q
o
F
.
Theorem 4.2. Let
GB be the ge
gi
nerator matrix of B
ven in Figure 2. Thn

GB


has a generator m-
trix that is permutae matrix given in
Figure 3.
e
on-equi
a
ti hvalent to t

 



1
1
1
1
2
2
1,21,31, 4
1,2 1,34
1,2 1,34
1,21,31, 4
2,4
2,3
,1 ,1
k q
k
k q
k q
k q
k
l
l
k
IAAA
vI vAvA
uI uAuA
vI
ujvD
uvI
 

 




 
 

 


3
,1 ,1
3,4
l
q
ll ll
kqq
uvD uvD
uvI uvD




 
 
 
 
 
 
 
 
 
 
 




1,
1,
q
vA
uA
uvA
vD


uvI uvAuvA


2,3
vD


ll
Figure 3. Generator Matrix of





2, 4q
uvD
ujvD
 
 
 


l
k
uvI uvD
ujvI


B
.
Copyright © 2013 SciRes. OJAppS
J. D. PALACIO, V. P. SISON
30
Proof: Let B have a generator matrix given in Figure
2. Then for every c can be expressed as yG
where , that is, z where
cB,
k
q
yR,
4
1
i
i
kk
1
k
ii
i
cs
iq
s
R and the are the k rows of '
i
zs
GB. Using
sis of Rq, cfurther be written
Now,
Hence,
any ba can
4444
,, ,,,
111111 11
kkkk
ijl iijiijiiji
ijijij ij
azb uzcvzduvz
 



 
 
44
,,
11 11
44
,,
11 11
.
kk
ij iiji
ij ij
kk
ij iiji
ij ij
cazbuz
cvz duvz


 
 


 
 
  
,,,
iii
Szuzvzuv
 
er
whenever or
for some
1,2,,
i
zi k
spans

B
. But
0
i
vz whenev112
1,, k k or ik
3q1,, k; ikk
0uz
i12 3
31, ,
q
ikk k
 ;
0 whenever ; and
12
1, ,ik kkkk 
i
uvz 1
ik
ii
uz jvzq
jF
i
r some l
whenever
k fo
Define the set
1ll
i
11
1, ,
ii
ik
.
sted a
m
as the resulting set once the unde-
sirable cases libove are deducted from the set S.
Notice that the eleents of
are the rows of the ma-
trix given in Fi we will denote by M. Now, define
as the matrix that consists of the rows
l
k
of M so that M can be
for all i.
Consider a ropressed as
a linear com any ,
t
gure 3
l
1
1
42 1,,
l
kk

B
1
22
42
ii
ii
k


written in the form 2
1
B
B
.
3q

ewish to show that te
B
W hrows of M are linearly inde-
pendent. Without loss of generality, let i
k
w of i
B. Clearly, it cannot
bination of rows from
1
be ex
of the '
j
Bs
ji. We know t

ha

1,, ,uvuv

are line-
arly independent and so any nonzero linear combination
of these vectors is not the zero vector. Thus, any row of
i
B cannot be written as a linear combination of rows of
any of the '
j
Bs, ji. Hence, the rows of M are line-
y independent. arl
The succeeding theorems aof
Theorem 4.2.
Corollary 4.3. If B is free with rank k, then
re direct consequences
B
is
free with rank 4k.
Corollary 4.4. Let B be a free rate-kn linear block
code overwith generator matrix 2
R

I
A
B
, then the
generator mtrix of the q-ary image of with respect to
the basis
a
1,1 ,1uvuv vuvuv,1uuv
  
uivalent to
  is
permutation-e
where
q
0
000
00 0
000 0
kkk
kk
kk
kk
IIIEFHDEDFDH
IIDE DE
IIDF DF
IID D
 






A
DEuFvHuv
 .
5. Distance Bounds of the Images of Linear
Block Codes over qqqq
F
uF vF uvF
um distance ofle indica-
tion of the goodness of a code. A field code can correct at
The minim a code gives a simp
most 1
2
errors where
is its minimum Hamming
distance. Hence, we are interested with upper bounds of
the minimum Hamming distance of the images of the
linear block codes over q
R. For the succeeding discus-
sions, we let B be a rate-kn linear block over q
R.
Also, we denote by
code
the minimum Hamming distance
of
B
.
eorem 5.1. (Singl B be free.
Then
Th eton-type Bound) Let
41.nk
 (3)
The above theorem is a direct consequence of Corol-
lary 4.3 and the Sinton Bound for codes over fields
wh
gle
f
ile the next theorem is a direct consequence of the
Plotkin Bound for codes over fields.
Theorem 5.2. (Plotkin-type Bound) Let B beree.
Then

41
414 .
1
k
k
qqn
q

(4)
The next bound is in terms of the average homogene-
ous weight
on q
F
andis-
tance of B.
the minimum Hamming d
Theorem 5.3. (Rains-type bound) For a code B,
4.
H
H
dd
(4
Proof: Note that
)
is bounded above by 4n. If for
every
,
H
H
x
Bw Bd then 4
H
d
. Now,
is
bounded below by
H
d
ng wei
si
of the Hammig
nce nimumro
value ht on
1 is the mi
nonze
q
F
. Thus, inequality (4)
pt of subcodes of B generated
holds.
Now, we use the conce
by x as defined by V. Sison and P. Sole in [4]. The sub-
code of B generated by
x
B
, denoted by
x
B, is the set
Copyright © 2013 SciRes. OJAppS
J. D. PALACIO, V. P. SISON 31

ax aR. A generalization of the Rabizzoni bound
was din [4]. Here wprove a parallel bound for
linear odes over q
Rhe proof presented here is
erived e
block cT.
based on the proof in [4].
Lemma 5.4. Let ,0xBx.
x
B is free if and only
if 4
x
Bq.
Proof:

Let
x
B be then the equation 0axfree
s
only the trivial solution. In particular,

0abx
aabplies ax bx. Thus,
ha
imb, that is, 4
x
Bq
.
Let 4
x
Bq. Then for any nonzero a and b,
ab axbx . That is,

0abxab
. But x
generates
x
B by definition. So,
x
B
t conse
is free.
quence of the car-The next statement is a
di
direc
nality of the ideals of q
R
Corollary 5.5. Let
x
B. Then

\0
nn
xuv if and only if m
x
Bp;

\
nn
x
ujv uv or
  
\
nn
x
vuv if and only
if 2m
x
Bp;

,\
n
x
uv S if and only if 3m
x
Bp where

q
jF
ujvv
.
Theorem 5.5. (Rbizzoni-type Bound) Let x be a
minimum H
nn
S
a
amming weight codeword. Then
14.
1
x
xH
d
(5)
x
Bq
More
Bq


over, if
x
B is free, then

3
41 .
1
xH
qqd
q




(6) 4

Proof: Let x be a minimum Hamming weight code-
word in B then consider subcode
x
B. Let
x
denote the
minimum Hamming distance of
x
B
. The minimum
Hamming distance of
x
B is still
H
d since
x
B is a
subcode of B. Also

x
B
is a subcode of
B
with
x
. The effective length of

x
B
is 4
H
d coming
from the
H
d nonze in. Direct application
of the Rabizzoni bou inuality (5) holds. By
Le i
ro posi
resu
t
nd l
ion
ts t
4, nlit
Conse free ra
s
o
x
eq
mma 5.equay (6) follows.
6. Example
ider thte-14 self-orthogonal code B over
2

11G uv . If

GIA then

1,11,0
k
ID E
R
F
generated by
HB is 4nary image of B was
ned with
1. Comparison of bounds for
1 1vuvu 
11 1
ht 0,4 or 8. The minim
,
um

110 and

001H. A codeword in B
either has homogeneous weig
amming distance of . The bi
obtai respect to the basis

1,1,1,1u v uvvu uvu uv v .
Table .
Singleton-type 813

Plotkin-type 88 8.53



4816
Rains-type
Rabizzoni-type
16
x
B
88
4
x
B
8 10.6
2
x
B
816
Using Corollary 4.4,
i
equivalent to
as a m dista
8 and le 1, w se
REFERENCES
. K. Gupta and
[2] T. Honold, “A Characterization of Finite Frobenius
Rings,” Ar Mathematik, V

GB
s permutation-
0111100100001110
1010100000111011
.
1100001111 0110
1001111 00000111




The image code hinimum Hammingnce of
is self-orthogonal. In Tab e cane that B
meets the upper bound of the Plotkin-type and Rabiz-
zoni-type bound.
00
0


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