Best Known (21, 29, s)-Nets in Base 256
(21, 29, 2129919)-Net over F256 — Constructive and digital
Digital (21, 29, 2129919)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 32769)-net over F256, using
- net defined by OOA [i] based on linear OOA(2567, 32769, F256, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [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(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(2567, 65538, F256, 4) (dual of [65538, 65531, 5]-code), using
- net defined by OOA [i] based on linear OOA(2567, 32769, F256, 4, 4) (dual of [(32769, 4), 131069, 5]-NRT-code), using
- digital (14, 22, 2097150)-net over F256, using
- net defined by OOA [i] based on linear OOA(25622, 2097150, F256, 8, 8) (dual of [(2097150, 8), 16777178, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(25622, 8388600, F256, 8) (dual of [8388600, 8388578, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(25622, large, F256, 8) (dual of [large, large−22, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(25622, 8388600, F256, 8) (dual of [8388600, 8388578, 9]-code), using
- net defined by OOA [i] based on linear OOA(25622, 2097150, F256, 8, 8) (dual of [(2097150, 8), 16777178, 9]-NRT-code), using
- digital (3, 7, 32769)-net over F256, using
(21, 29, large)-Net over F256 — Digital
Digital (21, 29, large)-net over F256, using
- 2 times m-reduction [i] based on digital (21, 31, large)-net over F256, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25631, large, F256, 10) (dual of [large, large−31, 11]-code), using
- strength reduction [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- strength reduction [i] based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(25631, large, F256, 10) (dual of [large, large−31, 11]-code), using
(21, 29, large)-Net in Base 256 — Upper bound on s
There is no (21, 29, large)-net in base 256, because
- 6 times m-reduction [i] would yield (21, 23, large)-net in base 256, but