ISIT2001, Washington, DC, June 24-29,2001
Matrix Characterization of Generalized Hamming Weights
G. Viswanath
B. Sundar Rajan’
Dept. of Elect. Comm. Engg.
Indian Institute of Science
Bangalore 560012, India
e-mail: gviswaaprotocol .
ece.iisc.ernet.in
Dept. of Elect. Comm. Engg.
Indian Institute of Science
Bangalore 560012, India
e-mail: bsrajanaece. i i s c .
ernet.i n
Abstract - A linear code with the systematic generator matrix [I I P ] is Maximum Distance Separable
(MDS) if and only if every square submatrix of P is
nonsingular. In this paper we obtain similar matrix
characterization for all linear codes with a specified
Hamming Weight Hierarchy (HWH). Using this
we characterize Near-MDS codes (NMDS), NearNear-MDS ( N ’ M D S ) codes and a generalization of
these codes called N p M D S codes in terms of their
systematic generator matrices.
0
1. a n d i < g 5 min{di-l,k), every ( g - 2 i + l + p I g )
sub matrix has rank 2 (g - i 1).
+
2. there exist a 9, i < g 5 m i n { d i , k } , such that
(9 - 2i + p , 9) sub matrix has rank equal t o (g - i ) .
0
Corollary 2: (N’MDS code) For N 2 M D S codes the systematic generator matrix characterization is
The support of a linear code C is the set of coordinate positions, where not all codewords of C are zero. The r-th generalized Hamming weight d,, 1 5 r 5 k , of a [ n , k ] linear code
C over a field is defined as the cardinality of the minimal support of an [n,r] subcode of C. The sequence ( d l , &, . . . ,&) is
called the Hamming Weight Hierarchy (HWH) of C [3]. The
relation d, = n - k r for r = 1 , 2 , . . . ,k characterizes MDS
codes.
NMDS Codes: [l]The class of [n,k ] codes where d l ( C ) =
( n - k ) , and d i ( C ) is ( n - k
i ) for i = 2 , 3 , . . . , k .
N’MDS Codes: [2] The class of [n,k ] codes where d l ( C ) =
( n - k - l),& ( C ) = ( n - k
1) and di(C) is ( n - k i ) for
i = 3 , 4 , . . . ]IC.
We generalize NMDS and N’MDS codes as
N p M D S Codes: The class of [ n , k ]codes where di(C) =
(n- k
2i - p - 1) for i = 1 , 2 , . . . ,k .
1. For 1 < g 5 min{(n - k - 2), k } every (9
matrix has rank 2 g
3. For 1 < g 5 min{(n - k - l),k } there exits a (g
sub matrix with rank (9 - 1).
+
+
5 . For 1 < g 5 min{(n - k ) , ( k - 2)) every (g,g
matrix has rank 9.
+
+
0
0
+ +
0
0
For i < g 5 min{di - 1,k } , every (9 +vi
matrix of P has rank 2 ( g - i + 1).
+ 1 - i ,9) sub
0
For 1 < g 5 min{(n - k - l),k } every (9
matrix has rank 2 g
+ 1,g) sub
For 1 < g 5 min{(n - k ) , k } there exists a (g,g) sub
matrix with rank equal to (9 - 1)
For 1 < g 5 min{(n- k ) , ( k - 1)) every (g,g
matrix has rank 9.
+ 1) sub
Corollary 3.1: If n > ( k + q ) the systematic generator matrix
of a N M D S code is characterized by every (g+l, 9) submatrix
having rank 2 g.
There exists a g, i < g 5 min{di,k}, such that rank of
(g - i vi,9) sub matrix is (9 - i)
+
REFERENCES
F~~1 < 5 min{(n - k ) , ( k - cl)} every (9,9 + p ) sub
matrix has rank 9.
For N’MDS codes the defect1 ‘%I for 1 5 i 5 PI is ( p + - i).
The defect is zero for all i > p .
Corollary 1: ( N p M D S codes) The systematic G matrix
characterization of N p M D S codes is as follows:
‘This work was partly supported by DST, India, through Research Grant No:III.5(31)/99-ET to B. S. Rajan
0-7803-7123-2/01/$I
0.00 02001 I EEE
+ 2) sub
Corollary 2.1: For k > q > 3 and n > 2q - 1 k the systematic generator matrix of a N 2M D S code is characterized
by every (g 2, g) submatrix having rank 1 g.
Corollary 3: ( N M D S code) Systematic generator matrix
characterization of N M D S codes is:
11. MATRIXCHARACTERIZATION
OF HWH
The M D S discrepancy of an [n,k] code is defined as the smallest p such that d,+l = n - k p 1.
Theorem 1: The systematic parity check matrix of an [n k ]
linear code with M D S discrepancy p and & ( C )= n-k+i-r/i
can be characterized as follows:
+ 1,g)
4. For 2 < g 5 min{(n - k l),k } there exits a (g - 1,g)
sub matrix with rank (g - 2).
+
+
+ 2,g) sub
2. For 2 < g 5 min{ ( n- k ) , k } every (g,g) sub matrix has
rank 2 (9 - 1)
+
+
F o r i = ( p + l ) , ... , k
1. For 1 < g 5 m i n { ( n - k ) , ( k - p ) } every ( g , g + p )
sub matrix has rank 9.
I. PRELIMINARIES
‘.
For i = 1,2, . . . ,p
61
[I] S. D. Dodunekov and I. N. Landgev, ’ On Near-MDS Codes’,
Technical Report, No:Lath-ISY-R-1563, Department of Electrical Engineerinn,
- Linkoping
.
-University, February, 1994.
[2] J. Olsson, ’ On Near-Near-MDS Codes’, Proceedings of Algebraic and Combinatorial Coding Theory Workshop, June, 1996
[3] V. K. wei, ”Generalized Hamming Weights for linear
codes,” IEEE Trans. on Information Theory, IT-Vo1.37, No:5,
Sept.1991, pp.1412-1418.
In