Best Known (131−32, 131, s)-Nets in Base 3
(131−32, 131, 328)-Net over F3 — Constructive and digital
Digital (99, 131, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (99, 132, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 33, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 33, 82)-net over F81, using
(131−32, 131, 677)-Net over F3 — Digital
Digital (99, 131, 677)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3131, 677, F3, 32) (dual of [677, 546, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3131, 756, F3, 32) (dual of [756, 625, 33]-code), using
- construction XX applied to C1 = C([334,364]), C2 = C([339,365]), C3 = C1 + C2 = C([339,364]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,364}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3109, 728, F3, 27) (dual of [728, 619, 28]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {339,340,…,365}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,365}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {339,340,…,364}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(37, 22, F3, 4) (dual of [22, 15, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- 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
- construction XX applied to C1 = C([334,364]), C2 = C([339,365]), C3 = C1 + C2 = C([339,364]), and C∩ = C1 ∩ C2 = C([334,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3131, 756, F3, 32) (dual of [756, 625, 33]-code), using
(131−32, 131, 27396)-Net in Base 3 — Upper bound on s
There is no (99, 131, 27397)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 318 447410 422916 917331 887298 946143 680789 425348 284600 278073 353665 > 3131 [i]