Best Known (89, 89+23, s)-Nets in Base 16
(89, 89+23, 95327)-Net over F16 — Constructive and digital
Digital (89, 112, 95327)-net over F16, using
- 162 times duplication [i] based on digital (87, 110, 95327)-net over F16, using
- net defined by OOA [i] based on linear OOA(16110, 95327, F16, 23, 23) (dual of [(95327, 23), 2192411, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16110, 1048598, F16, 23) (dual of [1048598, 1048488, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(16110, 1048600, F16, 23) (dual of [1048600, 1048490, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(1686, 1048576, F16, 19) (dual of [1048576, 1048490, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(16110, 1048600, F16, 23) (dual of [1048600, 1048490, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16110, 1048598, F16, 23) (dual of [1048598, 1048488, 24]-code), using
- net defined by OOA [i] based on linear OOA(16110, 95327, F16, 23, 23) (dual of [(95327, 23), 2192411, 24]-NRT-code), using
(89, 89+23, 1048607)-Net over F16 — Digital
Digital (89, 112, 1048607)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16112, 1048607, F16, 23) (dual of [1048607, 1048495, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(16106, 1048576, F16, 23) (dual of [1048576, 1048470, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
(89, 89+23, large)-Net in Base 16 — Upper bound on s
There is no (89, 112, large)-net in base 16, because
- 21 times m-reduction [i] would yield (89, 91, large)-net in base 16, but