Best Known (15, 15+8, s)-Nets in Base 3
(15, 15+8, 56)-Net over F3 — Constructive and digital
Digital (15, 23, 56)-net over F3, using
- 1 times m-reduction [i] based on digital (15, 24, 56)-net over F3, using
- trace code for nets [i] based on digital (3, 12, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- trace code for nets [i] based on digital (3, 12, 28)-net over F9, using
(15, 15+8, 80)-Net over F3 — Digital
Digital (15, 23, 80)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(323, 80, F3, 8) (dual of [80, 57, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(323, 88, F3, 8) (dual of [88, 65, 9]-code), using
- construction XX applied to Ce(7) ⊂ Ce(6) ⊂ Ce(4) [i] based on
- 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(313, 81, F3, 5) (dual of [81, 68, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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(7) ⊂ Ce(6) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(323, 88, F3, 8) (dual of [88, 65, 9]-code), using
(15, 15+8, 609)-Net in Base 3 — Upper bound on s
There is no (15, 23, 610)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 94437 264681 > 323 [i]