Best Known (99−25, 99, s)-Nets in Base 3
(99−25, 99, 252)-Net over F3 — Constructive and digital
Digital (74, 99, 252)-net over F3, using
- trace code for nets [i] based on digital (8, 33, 84)-net over F27, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 8 and N(F) ≥ 84, using
- net from sequence [i] based on digital (8, 83)-sequence over F27, using
(99−25, 99, 489)-Net over F3 — Digital
Digital (74, 99, 489)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(399, 489, F3, 25) (dual of [489, 390, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(399, 738, F3, 25) (dual of [738, 639, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(399, 738, F3, 25) (dual of [738, 639, 26]-code), using
(99−25, 99, 20824)-Net in Base 3 — Upper bound on s
There is no (74, 99, 20825)-net in base 3, because
- 1 times m-reduction [i] would yield (74, 98, 20825)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 57268 995125 201816 626639 372223 000592 703644 219441 > 398 [i]