Best Known (123−29, 123, s)-Nets in Base 3
(123−29, 123, 400)-Net over F3 — Constructive and digital
Digital (94, 123, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (94, 124, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 31, 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, 31, 100)-net over F81, using
(123−29, 123, 758)-Net over F3 — Digital
Digital (94, 123, 758)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 758, F3, 29) (dual of [758, 635, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3123, 762, F3, 29) (dual of [762, 639, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(22) ⊂ Ce(21) [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(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(310, 32, F3, 5) (dual of [32, 22, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(28) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3123, 762, F3, 29) (dual of [762, 639, 30]-code), using
(123−29, 123, 43457)-Net in Base 3 — Upper bound on s
There is no (94, 123, 43458)-net in base 3, because
- 1 times m-reduction [i] would yield (94, 122, 43458)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16176 184984 262501 801783 760909 614394 895402 495702 860726 379789 > 3122 [i]