Best Known (84, 84+30, s)-Nets in Base 16
(84, 84+30, 4369)-Net over F16 — Constructive and digital
Digital (84, 114, 4369)-net over F16, using
- 161 times duplication [i] based on digital (83, 113, 4369)-net over F16, using
- net defined by OOA [i] based on linear OOA(16113, 4369, F16, 30, 30) (dual of [(4369, 30), 130957, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(16113, 65535, F16, 30) (dual of [65535, 65422, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16113, 65536, F16, 30) (dual of [65536, 65423, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(16113, 65536, F16, 30) (dual of [65536, 65423, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(16113, 65535, F16, 30) (dual of [65535, 65422, 31]-code), using
- net defined by OOA [i] based on linear OOA(16113, 4369, F16, 30, 30) (dual of [(4369, 30), 130957, 31]-NRT-code), using
(84, 84+30, 54485)-Net over F16 — Digital
Digital (84, 114, 54485)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16114, 54485, F16, 30) (dual of [54485, 54371, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(16114, 65545, F16, 30) (dual of [65545, 65431, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(16113, 65536, F16, 30) (dual of [65536, 65423, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(16105, 65536, F16, 28) (dual of [65536, 65431, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(29) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(16114, 65545, F16, 30) (dual of [65545, 65431, 31]-code), using
(84, 84+30, large)-Net in Base 16 — Upper bound on s
There is no (84, 114, large)-net in base 16, because
- 28 times m-reduction [i] would yield (84, 86, large)-net in base 16, but