Best Known (84, 84+22, s)-Nets in Base 16
(84, 84+22, 95327)-Net over F16 — Constructive and digital
Digital (84, 106, 95327)-net over F16, using
- 161 times duplication [i] based on digital (83, 105, 95327)-net over F16, using
- net defined by OOA [i] based on linear OOA(16105, 95327, F16, 22, 22) (dual of [(95327, 22), 2097089, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(16105, 1048597, F16, 22) (dual of [1048597, 1048492, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(16105, 1048600, F16, 22) (dual of [1048600, 1048495, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(164, 24, F16, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,16)), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(16105, 1048600, F16, 22) (dual of [1048600, 1048495, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(16105, 1048597, F16, 22) (dual of [1048597, 1048492, 23]-code), using
- net defined by OOA [i] based on linear OOA(16105, 95327, F16, 22, 22) (dual of [(95327, 22), 2097089, 23]-NRT-code), using
(84, 84+22, 1048602)-Net over F16 — Digital
Digital (84, 106, 1048602)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16106, 1048602, F16, 22) (dual of [1048602, 1048496, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1681, 1048576, F16, 18) (dual of [1048576, 1048495, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(164, 25, F16, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,16)), using
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
(84, 84+22, large)-Net in Base 16 — Upper bound on s
There is no (84, 106, large)-net in base 16, because
- 20 times m-reduction [i] would yield (84, 86, large)-net in base 16, but