Best Known (132−31, 132, s)-Nets in Base 3
(132−31, 132, 464)-Net over F3 — Constructive and digital
Digital (101, 132, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 33, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
(132−31, 132, 803)-Net over F3 — Digital
Digital (101, 132, 803)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3132, 803, F3, 31) (dual of [803, 671, 32]-code), using
- 54 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0) [i] based on linear OA(3119, 736, F3, 31) (dual of [736, 617, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- 54 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0) [i] based on linear OA(3119, 736, F3, 31) (dual of [736, 617, 32]-code), using
(132−31, 132, 47147)-Net in Base 3 — Upper bound on s
There is no (101, 132, 47148)-net in base 3, because
- 1 times m-reduction [i] would yield (101, 131, 47148)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 318 348156 323272 424290 345066 442303 579921 857749 523261 918844 099601 > 3131 [i]