Best Known (116, 116+37, s)-Nets in Base 3
(116, 116+37, 400)-Net over F3 — Constructive and digital
Digital (116, 153, 400)-net over F3, using
- 31 times duplication [i] based on digital (115, 152, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 38, 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, 38, 100)-net over F81, using
(116, 116+37, 790)-Net over F3 — Digital
Digital (116, 153, 790)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3153, 790, F3, 37) (dual of [790, 637, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 791, F3, 37) (dual of [791, 638, 38]-code), using
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0) [i] based on linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 8 times 0, 1, 10 times 0, 1, 13 times 0) [i] based on linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 791, F3, 37) (dual of [791, 638, 38]-code), using
(116, 116+37, 40358)-Net in Base 3 — Upper bound on s
There is no (116, 153, 40359)-net in base 3, because
- 1 times m-reduction [i] would yield (116, 152, 40359)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 330537 621691 077258 649163 328645 610832 520340 458720 946560 999995 364324 922285 > 3152 [i]