Best Known (72, 72+26, s)-Nets in Base 16
(72, 72+26, 5041)-Net over F16 — Constructive and digital
Digital (72, 98, 5041)-net over F16, using
- 161 times duplication [i] based on digital (71, 97, 5041)-net over F16, using
- net defined by OOA [i] based on linear OOA(1697, 5041, F16, 26, 26) (dual of [(5041, 26), 130969, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(1697, 65533, F16, 26) (dual of [65533, 65436, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(1697, 65536, F16, 26) (dual of [65536, 65439, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(1697, 65536, F16, 26) (dual of [65536, 65439, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(1697, 65533, F16, 26) (dual of [65533, 65436, 27]-code), using
- net defined by OOA [i] based on linear OOA(1697, 5041, F16, 26, 26) (dual of [(5041, 26), 130969, 27]-NRT-code), using
(72, 72+26, 48063)-Net over F16 — Digital
Digital (72, 98, 48063)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1698, 48063, F16, 26) (dual of [48063, 47965, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 65545, F16, 26) (dual of [65545, 65447, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- linear OA(1697, 65536, F16, 26) (dual of [65536, 65439, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1689, 65536, F16, 24) (dual of [65536, 65447, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(161, 9, F16, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(1698, 65545, F16, 26) (dual of [65545, 65447, 27]-code), using
(72, 72+26, large)-Net in Base 16 — Upper bound on s
There is no (72, 98, large)-net in base 16, because
- 24 times m-reduction [i] would yield (72, 74, large)-net in base 16, but