Best Known (27, 38, s)-Nets in Base 3
(27, 38, 114)-Net over F3 — Constructive and digital
Digital (27, 38, 114)-net over F3, using
- 1 times m-reduction [i] based on digital (27, 39, 114)-net over F3, using
- trace code for nets [i] based on digital (1, 13, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- trace code for nets [i] based on digital (1, 13, 38)-net over F27, using
(27, 38, 183)-Net over F3 — Digital
Digital (27, 38, 183)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(338, 183, F3, 11) (dual of [183, 145, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(338, 251, F3, 11) (dual of [251, 213, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- linear OA(336, 243, F3, 11) (dual of [243, 207, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(331, 243, F3, 10) (dual of [243, 212, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(326, 243, F3, 8) (dual of [243, 217, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(10) ⊂ Ce(9) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(338, 251, F3, 11) (dual of [251, 213, 12]-code), using
(27, 38, 4416)-Net in Base 3 — Upper bound on s
There is no (27, 38, 4417)-net in base 3, because
- 1 times m-reduction [i] would yield (27, 37, 4417)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 450625 677768 391539 > 337 [i]