Best Known (51, 51+29, s)-Nets in Base 9
(51, 51+29, 344)-Net over F9 — Constructive and digital
Digital (51, 80, 344)-net over F9, using
- 8 times m-reduction [i] based on digital (51, 88, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 44, 172)-net over F81, using
(51, 51+29, 781)-Net over F9 — Digital
Digital (51, 80, 781)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(980, 781, F9, 29) (dual of [781, 701, 30]-code), using
- 45 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 28 times 0) [i] based on linear OA(976, 732, F9, 29) (dual of [732, 656, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(976, 729, F9, 29) (dual of [729, 653, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(973, 729, F9, 28) (dual of [729, 656, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(90, 3, F9, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 45 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0, 1, 11 times 0, 1, 28 times 0) [i] based on linear OA(976, 732, F9, 29) (dual of [732, 656, 30]-code), using
(51, 51+29, 183232)-Net in Base 9 — Upper bound on s
There is no (51, 80, 183233)-net in base 9, because
- 1 times m-reduction [i] would yield (51, 79, 183233)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 2427 598731 268242 545555 325018 964915 738106 047811 591967 321087 561128 563883 154225 > 979 [i]