Best Known (30, 55, s)-Nets in Base 256
(30, 55, 5462)-Net over F256 — Constructive and digital
Digital (30, 55, 5462)-net over F256, using
- 2563 times duplication [i] based on digital (27, 52, 5462)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25652, 65548, F256, 25) (dual of [65548, 65496, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25652, 65545, F256, 25) (dual of [65545, 65493, 26]-code), using
- net defined by OOA [i] based on linear OOA(25652, 5462, F256, 25, 25) (dual of [(5462, 25), 136498, 26]-NRT-code), using
(30, 55, 22480)-Net over F256 — Digital
Digital (30, 55, 22480)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25655, 22480, F256, 2, 25) (dual of [(22480, 2), 44905, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25655, 32778, F256, 2, 25) (dual of [(32778, 2), 65501, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25655, 65556, F256, 25) (dual of [65556, 65501, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(25635, 65536, F256, 18) (dual of [65536, 65501, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2566, 20, F256, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,256)), using
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- Reed–Solomon code RS(250,256) [i]
- discarding factors / shortening the dual code based on linear OA(2566, 256, F256, 6) (dual of [256, 250, 7]-code or 256-arc in PG(5,256)), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(25655, 65556, F256, 25) (dual of [65556, 65501, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(25655, 32778, F256, 2, 25) (dual of [(32778, 2), 65501, 26]-NRT-code), using
(30, 55, large)-Net in Base 256 — Upper bound on s
There is no (30, 55, large)-net in base 256, because
- 23 times m-reduction [i] would yield (30, 32, large)-net in base 256, but