Best Known (98, 129, s)-Nets in Base 3
(98, 129, 400)-Net over F3 — Constructive and digital
Digital (98, 129, 400)-net over F3, using
- 31 times duplication [i] based on digital (97, 128, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 32, 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, 32, 100)-net over F81, using
(98, 129, 719)-Net over F3 — Digital
Digital (98, 129, 719)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3129, 719, F3, 31) (dual of [719, 590, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3129, 758, F3, 31) (dual of [758, 629, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3121, 730, F3, 31) (dual of [730, 609, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(397, 730, F3, 25) (dual of [730, 633, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3129, 758, F3, 31) (dual of [758, 629, 32]-code), using
(98, 129, 37844)-Net in Base 3 — Upper bound on s
There is no (98, 129, 37845)-net in base 3, because
- 1 times m-reduction [i] would yield (98, 128, 37845)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 11 791608 037427 732204 629455 678009 564286 326886 316751 690870 846395 > 3128 [i]