Best Known (82−19, 82, s)-Nets in Base 3
(82−19, 82, 400)-Net over F3 — Constructive and digital
Digital (63, 82, 400)-net over F3, using
- 32 times duplication [i] based on digital (61, 80, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
(82−19, 82, 658)-Net over F3 — Digital
Digital (63, 82, 658)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(382, 658, F3, 19) (dual of [658, 576, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 740, F3, 19) (dual of [740, 658, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([1,9]) [i] based on
- linear OA(373, 730, F3, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(372, 730, F3, 9) (dual of [730, 658, 10]-code), using the narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- construction X applied to C([0,9]) ⊂ C([1,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(382, 740, F3, 19) (dual of [740, 658, 20]-code), using
(82−19, 82, 40805)-Net in Base 3 — Upper bound on s
There is no (63, 82, 40806)-net in base 3, because
- 1 times m-reduction [i] would yield (63, 81, 40806)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 443 442364 829758 297611 528800 800363 649085 > 381 [i]