A new code capable of correcting a single asymmetric error is proposed. This code is of length 19 and has 28142 codewords, improving the best known code by 110 code words

Al- Bassam, Sulaiman

Al-Muhammadi, Sultan

King Fahd University of Petroleum and Minerals

A NEW SINGLE ASYMMETRIC ERROR CORRECTING CODE OF 
LENGTH 19 
Sultan Al-Muhammadi and Sulaiman Al- Bassam 
Information and Computer Science Department 
King Fahad University of Petroleum & Minerals 
Dhahran - Saudi Arabia 
muhamadi(9dpc.kfupm.edu.sa 
ABSTRACT 
A new code capable of correcting a single asym- 
metric error is proposed. This code is of length 
19 and has 28142 codewords, improving the best 
known code by 110 code words. 
1. INTRODUCTION 
In this work, we present a new code capable of 
correcting a single asymmetric error. The proposed 
code is of asymmetric distance 2 and of length 19. 
The asymmetric distance of two binary vectors, z 
and y, of the same length is defined by 
da@,Y) = maz{N(z,Y),N(Y,z)} 
where N(z,y) = I{i : xi = landyi = O}l, i.e. the 
number of positions where x has a 1 and y has a 
0. The minimum asymmetric distance of a code C 
is defined as follows 
Da(C) - min{d,(z, y) : z, y E Candz # y} 
A code C can correct d asymmetric errors or 
fewer if Da(C) > d. The proposed code has asym- 
metric distance two so it is capable of correcting a 
single asymmetric error. 
2. CONSTRUCTION METHOD 
The construction of the proposed code (C) is based 
on the Cartesian product of two sets of partitioned 
codes, say {AI, A2,. . .} and {B1,B2,. . .}, where 
C = A1 x B1 U A2 x Bz U A3 x B3 U ... (1) 
These two sets are defined as follows: 
Let A be the set of all the 2 P  binary vectors 
of length p and let A I ,  Az, ~ - . , A,, be a partition 
of A, i.e. A; n Ai = q5 and U A; = A, such that 
D,(A;) 2 2 for 1 5 i 5 p l .  
Also, let B be the set of the 24-1 even weight 
binary vectors of length q and B1, B2, - - ~ , Bgl be a 
partition of B such that D,(Bj) 2 2 for 1 5 j 5 41. 
It was shown in [l] that the code constructed us- 
ing (1) above, is of asymmetric distance two. The 
length of the constructed code is p + q  and the car- 
dinality of the code is: 
Ex amp 1 e : 
To construct a single asymmetric error correct- 
ing code with n = 6, let p = 2 and q = 4. Then 
A = {00,01,10,11} can be partitioned into AI = 
{00,11},A2 = { O l } ,  and At = {lo}. And, B = 
(0000,0011,0101,. . . ,1111) can be partitioned into 
B1 = {0000,0011,1100,1111}, B2 = (0101, lolo}, 
and B3 = (0110,1001). 
We obtain a code C of length 6 where C = A1 x 
BlUA2 x B ~ U A ~  x B3 having 2*4+1*2+1*2 = 12 
code words as shown in Fig.1. 
CCECE ' 97 0-7803-3716-6 /97/$5.00 0 1997 IEEE 
00 0000 
00 0011 
00 1100 
00 1111 Ai x B1 
11 0000 
11 0011 
11 1100 
11 1111 
01 0101 
01 1010 A2 x B2 
10 0110 
10 1001 AB x B3 
Fig. 1: Constructing a single asymmet- 
ric error correcting code for n = 6. 
3. THE NEW CODE OF LENGTH 19 
The best known single asymmetric error correcting 
code of length 19 and 28032 code words was given 
in [l]. It was obtained from the Cartesian prod- 
uct of two partitions, A and B. The A partition 
contains all binary vectors of length p = 7 and it 
is obtained from the Abelian group 28 (see (41); 
yielding 8 partitions each of size 16 code words, 
i.e. A = { A I ,  Aa, ~ ~. Ag} where ]Ail = 16 for all 
i. The B partition conatins all the even binary 
vectors of length q E 12 and it has 11 partitions 
B z { B1, B2, ~ . ~ B1 } of the following eleven sizes: 
248, 246, 234, 234, 224, 198,192, 176,136,94, and 
66, respectively (see [3]) .  
Using the Cartesian product method with these 
A and B partitions, the code of length 19 and of 
size: 16 * (248 + 246 + 234 + 234 + 224 + 198 + 
192 + 176) + 0 * (136 + 94 + 66) = 28032 code 
words, was constructed in [l]. 
Here, we obtain a new A partition of length 7 
with 9 different partitions of the following sizes: 
18, 18, 16, 16, 16, 15, 14, 12, and 3. These parti- 
tions are shown in Fig.2. Applying the Cartesian 
product method using the new A partition with the 
above B partition gives a new code of length 19 and 
of size: 18*(248+246) +16*(234+234+224)+15* 
198+14*192+12*176+3*136+0*(94+66) = 28142 
code words. This code improves the best known 
code by 110 code words. 
76 
A1 
0000001 
0001010 
0100100 
1010000 
0010101 
0100011 
0111000 
1000110 
1001001 
0001111 
0110110 
1011010 
1101100 
1110001 
0111011 
1011101 
110011 1 
1111110 
-45 
0000010 
0000101 
0110000 
1001000 
0011100 
0101010 
1000011 
1100100 
0011011 
0100111 
1010101 
1101001 
0111101 
1101110 
1110011 
1011111 
A2 
0000100 
0010001 
0101000 
1000010 
0001101 
0011010 
0100110 
1010100 
1100001 
0110011 
0111100 
1000111 
1011001 
1101010 
00111 11 
1101101 
1110110 
1111011 
-46 
0001000 
0010010 
1000001 
0001 110 
0110001 
1011000 
1100010 
0010111 
0101101 
0111010 
1001011 
1110100 
1011110 
1111001 
011 1111 
A3 
0010000 
0100001 
1000100 
0000111 
0011001 
0110100 
1010010 
1101000 
0011110 
0101011 
1001101 
1100110 
0110111 
1011011 
111 1100 
1101111 
A7 
0100000 
0000011 
0011000 
0010110 
0101001 
1000101 
1001010 
1110000 
0110101 
1010011. 
1011100 
010111i 
1111010 
1110111 
A4 
0000000 
0010100 
1100000 
0001011 
0100101 
0110010 
1001100 
1010001 
0011101 
0101110 
1010110 
1100011 
11 11000 
1001111 
1110101 
1111111 
A8 
1000000 
0001100 
0100010 
0010011 
0111001 
1001110 
1100101 
1110010 
0111110 
1010111 
1101011 
1111101 
A9 
0000110 
0001001 
0101100 
Fig. 2: The new A partition of the 27 
binary vectors. 
The new A partition is obtained by graph-coloring 
method. In this method, we construct a graph 
G = ( N , E ) ,  where N is the set of nodes and E 
is the set of edges in the graph, as follows: 
N is the set of all 27 binary vectors and 
.E={(z,Y) : z , Y E N ; D a ( z , Y ) = l l  
77 
The nodes of the graph are colored using 9 col- 
ors, i.e. Vz, y E N ,  if (z, y) E E then z and y have 
different colors. Cearly, the set of all nodes having 
color k, say A k ,  satisfy Da(Ak) 2 2. 
4. ACKNOWLEDGEMENTS 
This work is supported by King Fahad University 
of Petroleum & Minerals. 
5. REFERENCES 
[l] S. Al-Bassam, et. al., “New single asymmet- 
ric error correcting codes” , IEEE Trans. on 
Inform. Theory, to appear. 
[2] Z. Zhang and X. Xia, “New lower bounds for 
binary codes of assymmetric distance two”, 
IEEE Dans. on Inform. Theory, Vol. IT- 
38, pp. 1592-1597, Sep. 1992. 
[3] A. E. Brouwer, et. al., “A new table of con- 
stant weight codes”, IEEE Rans. on Inform. 
Theory, Vol. IT-36, pp. 1334-1380, Nov. 
1990. 
[4] R. R. Varshamov, “A class of codes for asym- 
metric channels and a problem from the ad- 
ditive theory of numbers”, IEEE Trans. on 
Inform. Theory, Vol. IT-19, pp. 92-95, Jan. 
1973. 


A NEW SINGLE ASYMMETRIC ERROR CORRECTING CODE OF LENGTH 19

KFUPM ePrints

A NEW SINGLE ASYMMETRIC ERROR CORRECTING CODE OF LENGTH 19 Sultan Al-Muhammadi and Sulaiman Al- Bassam Information and Computer Science Department King Fahad University of Petroleum & Minerals Dhahran - Saudi Arabia muhamadi(9dpc.kfupm.edu.sa ABSTRACT A new code capable of correcting a single asym- metric error is proposed. This code is of length 19 and has 28142 codewords, improving the best known code by 110 code words. 1. INTRODUCTION In this work, we present a new code capable of correcting a single asymmetric error. The proposed code is of asymmetric distance 2 and of length 19. The asymmetric distance of two binary vectors, z and y, of the same length is defined by da@,Y) = maz{N(z,Y),N(Y,z)} where N(z,y) = I{i : xi = landyi = O}l, i.e. the number of positions where x has a 1 and y has a 0. The minimum asymmetric distance of a code C is defined as follows Da(C) - min{d,(z, y) : z, y E Candz # y} A code C can correct d asymmetric errors or fewer if Da(C) > d. The proposed code has asym- metric distance two so it is capable of correcting a single asymmetric error. 2. CONSTRUCTION METHOD The construction of the proposed code (C) is based on the Cartesian product of two sets of partitioned codes, say {AI, A2,. . .} and {B1,B2,. . .}, where C = A1 x B1 U A2 x Bz U A3 x B3 U ... (1) These two sets are defined as follows: Let A be the set of all the 2 P  binary vectors of length p and let A I ,  Az, ~ - . , A,, be a partition of A, i.e. A; n Ai = q5 and U A; = A, such that D,(A;) 2 2 for 1 5 i 5 p l .  Also, let B be the set of the 24-1 even weight binary vectors of length q and B1, B2, - - ~ , Bgl be a partition of B such that D,(Bj) 2 2 for 1 5 j 5 41. It was shown in [l] that the code constructed us- ing (1) above, is of asymmetric distance two. The length of the constructed code is p + q  and the car- dinality of the code is: Ex amp 1 e : To construct a single asymmetric error correct- ing code with n = 6, let p = 2 and q = 4. Then A = {00,01,10,11} can be partitioned into AI = {00,11},A2 = { O l } ,  and At = {lo}. And, B = (0000,0011,0101,. . . ,1111) can be partitioned into B1 = {0000,0011,1100,1111}, B2 = (0101, lolo}, and B3 = (0110,1001). We obtain a code C of length 6 where C = A1 x BlUA2 x B ~ U A ~  x B3 having 2*4+1*2+1*2 = 12 code words as shown in Fig.1. CCECE ' 97 0-7803-3716-6 /97/$5.00 0 1997 IEEE 00 0000 00 0011 00 1100 00 1111 Ai x B1 11 0000 11 0011 11 1100 11 1111 01 0101 01 1010 A2 x B2 10 0110 10 1001 AB x B3 Fig. 1: Constructing a single asymmet- ric error correcting code for n = 6. 3. THE NEW CODE OF LENGTH 19 The best known single asymmetric error correcting code of length 19 and 28032 code words was given in [l]. It was obtained from the Cartesian prod- uct of two partitions, A and B. The A partition contains all binary vectors of length p = 7 and it is obtained from the Abelian group 28 (see (41); yielding 8 partitions each of size 16 code words, i.e. A = { A I ,  Aa, ~ ~. Ag} where ]Ail = 16 for all i. The B partition conatins all the even binary vectors of length q E 12 and it has 11 partitions B z { B1, B2, ~ . ~ B1 } of the following eleven sizes: 248, 246, 234, 234, 224, 198,192, 176,136,94, and 66, respectively (see [3]) .  Using the Cartesian product method with these A and B partitions, the code of length 19 and of size: 16 * (248 + 246 + 234 + 234 + 224 + 198 + 192 + 176) + 0 * (136 + 94 + 66) = 28032 code words, was constructed in [l]. Here, we obtain a new A partition of length 7 with 9 different partitions of the following sizes: 18, 18, 16, 16, 16, 15, 14, 12, and 3. These parti- tions are shown in Fig.2. Applying the Cartesian product method using the new A partition with the above B partition gives a new code of length 19 and of size: 18*(248+246) +16*(234+234+224)+15* 198+14*192+12*176+3*136+0*(94+66) = 28142 code words. This code improves the best known code by 110 code words. 76 A1 0000001 0001010 0100100 1010000 0010101 0100011 0111000 1000110 1001001 0001111 0110110 1011010 1101100 1110001 0111011 1011101 110011 1 1111110 -45 0000010 0000101 0110000 1001000 0011100 0101010 1000011 1100100 0011011 0100111 1010101 1101001 0111101 1101110 1110011 1011111 A2 0000100 0010001 0101000 1000010 0001101 0011010 0100110 1010100 1100001 0110011 0111100 1000111 1011001 1101010 00111 11 1101101 1110110 1111011 -46 0001000 0010010 1000001 0001 110 0110001 1011000 1100010 0010111 0101101 0111010 1001011 1110100 1011110 1111001 011 1111 A3 0010000 0100001 1000100 0000111 0011001 0110100 1010010 1101000 0011110 0101011 1001101 1100110 0110111 1011011 111 1100 1101111 A7 0100000 0000011 0011000 0010110 0101001 1000101 1001010 1110000 0110101 1010011. 1011100 010111i 1111010 1110111 A4 0000000 0010100 1100000 0001011 0100101 0110010 1001100 1010001 0011101 0101110 1010110 1100011 11 11000 1001111 1110101 1111111 A8 1000000 0001100 0100010 0010011 0111001 1001110 1100101 1110010 0111110 1010111 1101011 1111101 A9 0000110 0001001 0101100 Fig. 2: The new A partition of the 27 binary vectors. The new A partition is obtained by graph-coloring method. In this method, we construct a graph G = ( N , E ) ,  where N is the set of nodes and E is the set of edges in the graph, as follows: N is the set of all 27 binary vectors and .E={(z,Y) : z , Y E N ; D a ( z , Y ) = l l  77 The nodes of the graph are colored using 9 col- ors, i.e. Vz, y E N ,  if (z, y) E E then z and y have different colors. Cearly, the set of all nodes having color k, say A k ,  satisfy Da(Ak) 2 2. 4. ACKNOWLEDGEMENTS This work is supported by King Fahad University of Petroleum & Minerals. 5. REFERENCES [l] S. Al-Bassam, et. al., “New single asymmet- ric error correcting codes” , IEEE Trans. on Inform. Theory, to appear. [2] Z. Zhang and X. Xia, “New lower bounds for binary codes of assymmetric distance two”, IEEE Dans. on Inform. Theory, Vol. IT- 38, pp. 1592-1597, Sep. 1992. [3] A. E. Brouwer, et. al., “A new table of con- stant weight codes”, IEEE Rans. on Inform. Theory, Vol. IT-36, pp. 1334-1380, Nov. 1990. [4] R. R. Varshamov, “A class of codes for asym- metric channels and a problem from the ad- ditive theory of numbers”, IEEE Trans. on Inform. Theory, Vol. IT-19, pp. 92-95, Jan. 1973. 

https://eprints.kfupm.edu.sa/id/eprint/10716/1/Coding_CCECE_97.pdf

A NEW SINGLE ASYMMETRIC ERROR CORRECTING CODE OF LENGTH 19

Abstract

Similar works

Full text

Available Versions

King Fahd University of Petroleum and Minerals

KFUPM ePrints