Best Known (59, 59+17, s)-Nets in Base 3
(59, 59+17, 464)-Net over F3 — Constructive and digital
Digital (59, 76, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 19, 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
(59, 59+17, 767)-Net over F3 — Digital
Digital (59, 76, 767)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(376, 767, F3, 17) (dual of [767, 691, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(376, 773, F3, 17) (dual of [773, 697, 18]-code), using
- 29 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(367, 735, F3, 17) (dual of [735, 668, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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(16) ⊂ Ce(15) [i] based on
- 29 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0) [i] based on linear OA(367, 735, F3, 17) (dual of [735, 668, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(376, 773, F3, 17) (dual of [773, 697, 18]-code), using
(59, 59+17, 55925)-Net in Base 3 — Upper bound on s
There is no (59, 76, 55926)-net in base 3, because
- 1 times m-reduction [i] would yield (59, 75, 55926)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 608268 578369 773975 357424 715407 509201 > 375 [i]