Best Known (31−11, 31, s)-Nets in Base 3
(31−11, 31, 60)-Net over F3 — Constructive and digital
Digital (20, 31, 60)-net over F3, using
- 1 times m-reduction [i] based on digital (20, 32, 60)-net over F3, using
- trace code for nets [i] based on digital (4, 16, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- trace code for nets [i] based on digital (4, 16, 30)-net over F9, using
(31−11, 31, 74)-Net over F3 — Digital
Digital (20, 31, 74)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(331, 74, F3, 11) (dual of [74, 43, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(331, 92, F3, 11) (dual of [92, 61, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- linear OA(327, 81, F3, 11) (dual of [81, 54, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(321, 81, F3, 8) (dual of [81, 60, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(317, 81, F3, 7) (dual of [81, 64, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(33, 10, F3, 2) (dual of [10, 7, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(331, 92, F3, 11) (dual of [92, 61, 12]-code), using
(31−11, 31, 945)-Net in Base 3 — Upper bound on s
There is no (20, 31, 946)-net in base 3, because
- 1 times m-reduction [i] would yield (20, 30, 946)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 206 875445 228541 > 330 [i]