Best Known (82−23, 82, s)-Nets in Base 3
(82−23, 82, 192)-Net over F3 — Constructive and digital
Digital (59, 82, 192)-net over F3, using
- 31 times duplication [i] based on digital (58, 81, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 27, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 27, 64)-net over F27, using
(82−23, 82, 283)-Net over F3 — Digital
Digital (59, 82, 283)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(382, 283, F3, 23) (dual of [283, 201, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 284, F3, 23) (dual of [284, 202, 24]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0) [i] based on linear OA(376, 248, F3, 23) (dual of [248, 172, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(376, 243, F3, 23) (dual of [243, 167, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(371, 243, F3, 22) (dual of [243, 172, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 9 times 0) [i] based on linear OA(376, 248, F3, 23) (dual of [248, 172, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(382, 284, F3, 23) (dual of [284, 202, 24]-code), using
(82−23, 82, 7993)-Net in Base 3 — Upper bound on s
There is no (59, 82, 7994)-net in base 3, because
- 1 times m-reduction [i] would yield (59, 81, 7994)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 443 452478 620832 241840 668381 479833 129633 > 381 [i]