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
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 .

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
1ima
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

 .

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.
 
\0xuvq 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  xx 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 
.

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
3q1,, k; ikk
0uz
i12 3
31, , q
ikk k

 ;
0 whenever ; and
12
1, ,ik kkkk 
i
uvz 1 ik
ii
uz jvzq 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

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
aabplies 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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
1v-Constacyclic Codes
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F
uF vFuvF,” Jour
stitute, Vol. 347, 2011, pp. 2652-
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

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2


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
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
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