Best Known (123−27, 123, s)-Nets in Base 3
(123−27, 123, 464)-Net over F3 — Constructive and digital
Digital (96, 123, 464)-net over F3, using
- t-expansion [i] based on digital (95, 123, 464)-net over F3, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (95, 124, 464)-net over F3, using
(123−27, 123, 967)-Net over F3 — Digital
Digital (96, 123, 967)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 967, F3, 27) (dual of [967, 844, 28]-code), using
- 224 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 37 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
- 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]
- 224 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 37 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
(123−27, 123, 85096)-Net in Base 3 — Upper bound on s
There is no (96, 123, 85097)-net in base 3, because
- 1 times m-reduction [i] would yield (96, 122, 85097)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16174 247519 467586 305422 157039 445875 388103 582527 111672 073891 > 3122 [i]