Best Known (149−26, 149, s)-Nets in Base 3
(149−26, 149, 700)-Net over F3 — Constructive and digital
Digital (123, 149, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 17, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (106, 132, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 33, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 33, 172)-net over F81, using
- digital (4, 17, 12)-net over F3, using
(149−26, 149, 4269)-Net over F3 — Digital
Digital (123, 149, 4269)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3149, 4269, F3, 26) (dual of [4269, 4120, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3149, 6605, F3, 26) (dual of [6605, 6456, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3149, 6605, F3, 26) (dual of [6605, 6456, 27]-code), using
(149−26, 149, 833514)-Net in Base 3 — Upper bound on s
There is no (123, 149, 833515)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 123330 006062 125409 551633 381454 879320 733048 281495 184606 826941 854259 710527 > 3149 [i]