Best Known (126, 126+33, s)-Nets in Base 3
(126, 126+33, 640)-Net over F3 — Constructive and digital
Digital (126, 159, 640)-net over F3, using
- t-expansion [i] based on digital (125, 159, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (125, 160, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 40, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 40, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (125, 160, 640)-net over F3, using
(126, 126+33, 1649)-Net over F3 — Digital
Digital (126, 159, 1649)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 1649, F3, 33) (dual of [1649, 1490, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 2207, F3, 33) (dual of [2207, 2048, 34]-code), using
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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(31, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(33) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 2207, F3, 33) (dual of [2207, 2048, 34]-code), using
(126, 126+33, 175001)-Net in Base 3 — Upper bound on s
There is no (126, 159, 175002)-net in base 3, because
- 1 times m-reduction [i] would yield (126, 158, 175002)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2427 526219 760931 930661 912139 114119 381888 021462 027018 372255 571750 647882 211745 > 3158 [i]