Best Known (37, 37+13, s)-Nets in Base 256
(37, 37+13, 1419948)-Net over F256 — Constructive and digital
Digital (37, 50, 1419948)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (7, 13, 21848)-net over F256, using
- net defined by OOA [i] based on linear OOA(25613, 21848, F256, 6, 6) (dual of [(21848, 6), 131075, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(25613, 65544, F256, 6) (dual of [65544, 65531, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2565, 65536, F256, 3) (dual of [65536, 65531, 4]-code or 65536-cap in PG(4,256)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(25613, 65544, F256, 6) (dual of [65544, 65531, 7]-code), using
- net defined by OOA [i] based on linear OOA(25613, 21848, F256, 6, 6) (dual of [(21848, 6), 131075, 7]-NRT-code), using
- digital (24, 37, 1398100)-net over F256, using
- net defined by OOA [i] based on linear OOA(25637, 1398100, F256, 13, 13) (dual of [(1398100, 13), 18175263, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25637, 8388601, F256, 13) (dual of [8388601, 8388564, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25637, 8388601, F256, 13) (dual of [8388601, 8388564, 14]-code), using
- net defined by OOA [i] based on linear OOA(25637, 1398100, F256, 13, 13) (dual of [(1398100, 13), 18175263, 14]-NRT-code), using
- digital (7, 13, 21848)-net over F256, using
(37, 37+13, large)-Net over F256 — Digital
Digital (37, 50, large)-net over F256, using
- t-expansion [i] based on digital (35, 50, large)-net over F256, using
- 1 times m-reduction [i] based on digital (35, 51, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25651, large, F256, 16) (dual of [large, large−51, 17]-code), using
- 5 times code embedding in larger space [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 5 times code embedding in larger space [i] based on linear OA(25646, large, F256, 16) (dual of [large, large−46, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25651, large, F256, 16) (dual of [large, large−51, 17]-code), using
- 1 times m-reduction [i] based on digital (35, 51, large)-net over F256, using
(37, 37+13, large)-Net in Base 256 — Upper bound on s
There is no (37, 50, large)-net in base 256, because
- 11 times m-reduction [i] would yield (37, 39, large)-net in base 256, but