Best Known (125−29, 125, s)-Nets in Base 3
(125−29, 125, 464)-Net over F3 — Constructive and digital
Digital (96, 125, 464)-net over F3, using
- 31 times duplication [i] based on digital (95, 124, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 31, 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
- trace code for nets [i] based on digital (2, 31, 116)-net over F81, using
(125−29, 125, 805)-Net over F3 — Digital
Digital (96, 125, 805)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 805, F3, 29) (dual of [805, 680, 30]-code), using
- 60 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0) [i] based on linear OA(3112, 732, F3, 29) (dual of [732, 620, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 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(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(30, 3, F3, 0) (dual of [3, 3, 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(28) ⊂ Ce(27) [i] based on
- 60 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 0, 0, 1, 0, 0, 1, 4 times 0, 1, 6 times 0, 1, 9 times 0, 1, 12 times 0, 1, 15 times 0) [i] based on linear OA(3112, 732, F3, 29) (dual of [732, 620, 30]-code), using
(125−29, 125, 50844)-Net in Base 3 — Upper bound on s
There is no (96, 125, 50845)-net in base 3, because
- 1 times m-reduction [i] would yield (96, 124, 50845)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 145579 782918 578708 555893 061540 096766 369384 301390 371903 843537 > 3124 [i]