Griesmer Bound
An [n,k,d]q code means a linear code of length n, dimension k and minimum Hamming distance d over the field of q elements GF(q).
nq(k,d) = the minimum length n for which an [n,k,d]q code exists.
The Griesmer bound says that an [n,k,d]q code satisfies
Linear Codes of Dimension k over GF(q) with k = 1 or 2
The Griesmer bound is attained for all d and q for k = 1, 2.
Binary Linear Codes
All values of n2(k,d) have been determined for k < 9, see
I. Bouyukliev, D.B. Jaffe, Z. Vabrek,
The smallest length of eight-dimensional binary linear codes with prescribed minimum distance,
IEEE Trans. Inform. Theory 46(4) (2000) 1539--1544.
Linear Codes of Dimension Three over GF(q), 2 < q <= 9
The Griesmer bound is attained for all d except the following, for which nq(3,d) = gq(3,d) + 1:
q=3; d=3
q=4; d=7, 8
q=5; d=5, 9, 10, 13, 14, 15
q=7; d=7, 13, 14, 19-21, 25-28, 31-35
q=8; d=13-16, 29-32, 37-40, 43-48
q=9; d=9, 15-18, 25-27, 33-36, 43-45, 49-54, 58-63.
See also Ball's table.
Ternary Linear Codes for k = 4, 5, 6
When k = 4, the Griesmer bound is attained for all d except the following 7 cases:
n3(4,d) = g3(4,d) + 1 for d = 3, 7-9, 13-15.
n3(5,d) table (The Griesmer bound is attained for the missing d in the table.)
Some 9-divisible ternary linear codes of dimensions 7, 8, 9, 10.
Quaternary Linear Codes for k = 4, 5
When k = 4, the Griesmer bound is attained for all d except the following 30 cases:
n4(4,d) = g4(4,d) + 1 for d = 3, 4, 7, 8, 13-16, 23-32, 37-44, 77-80.
n4(5,d) table (The Griesmer bound is attained for the missing d in the table.)
Linear Codes over GF(5) for k = 4, 5
When k = 4, the Griesmer bound is attained for all d except the following cases:
n5(4,d) = g5(4,d) + 1 for d = 4, 5, 9, 10, 12-15, 21-24, 31-50, 61-75, 83-95, 146-150, 163-175.
n5(4,d) = g5(4,d) + 2 for d = 25.
n5(4,d) = g5(4,d) or g5(4,d) + 1 for d = 81, 161. <--- only two open cases!
Note (2009):
The nonexistence of [g5(4,d),4,d]5 codes for d = 31,
32, 36, 37 was proved by
Edel
and Landjev.
Note (2015):
The nonexistence of [g5(4,d),4,d]5 codes for d = 82, 162 was announced by
Rousseva and Landjev at ALCOMA15.
n5(5,d) table (n5(5,d) = g5(5,d) for all d > 1375. )
Linear Codes over GF(7) for k = 4
n7(4,d) table (n7(4,d) = g7(4,d) for all d > 539. )
Linear Codes over GF(8) for k = 4
n8(4,d) table after On optimal linear codes over F8
Linear Codes over GF(9) for k = 4
n9(4,d) table (n9(4,d) = g9(4,d) for all d > 1215. )
See Markus Grassl's table for other cases.
If you get a new result improving the tables in this site, please inform it
to Tatsuya Maruta.