Best Known (24, 24+21, s)-Nets in Base 256
(24, 24+21, 6554)-Net over F256 — Constructive and digital
Digital (24, 45, 6554)-net over F256, using
- 2563 times duplication [i] based on digital (21, 42, 6554)-net over F256, using
- net defined by OOA [i] based on linear OOA(25642, 6554, F256, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25642, 65541, F256, 21) (dual of [65541, 65499, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25642, 65542, F256, 21) (dual of [65542, 65500, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(25642, 65541, F256, 21) (dual of [65541, 65499, 22]-code), using
- net defined by OOA [i] based on linear OOA(25642, 6554, F256, 21, 21) (dual of [(6554, 21), 137592, 22]-NRT-code), using
(24, 24+21, 21850)-Net over F256 — Digital
Digital (24, 45, 21850)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25645, 21850, F256, 3, 21) (dual of [(21850, 3), 65505, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25645, 65550, F256, 21) (dual of [65550, 65505, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(25641, 65536, F256, 21) (dual of [65536, 65495, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(25631, 65536, F256, 16) (dual of [65536, 65505, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2564, 14, F256, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,256)), using
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- Reed–Solomon code RS(252,256) [i]
- discarding factors / shortening the dual code based on linear OA(2564, 256, F256, 4) (dual of [256, 252, 5]-code or 256-arc in PG(3,256)), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- OOA 3-folding [i] based on linear OA(25645, 65550, F256, 21) (dual of [65550, 65505, 22]-code), using
(24, 24+21, large)-Net in Base 256 — Upper bound on s
There is no (24, 45, large)-net in base 256, because
- 19 times m-reduction [i] would yield (24, 26, large)-net in base 256, but