Best Known (67, 88, s)-Nets in Base 3
(67, 88, 400)-Net over F3 — Constructive and digital
Digital (67, 88, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 22, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(67, 88, 590)-Net over F3 — Digital
Digital (67, 88, 590)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(388, 590, F3, 21) (dual of [590, 502, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(388, 747, F3, 21) (dual of [747, 659, 22]-code), using
- construction XX applied to C1 = C([345,364]), C2 = C([348,365]), C3 = C1 + C2 = C([348,364]), and C∩ = C1 ∩ C2 = C([345,365]) [i] based on
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {345,346,…,364}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(373, 728, F3, 18) (dual of [728, 655, 19]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {348,349,…,365}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(385, 728, F3, 21) (dual of [728, 643, 22]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {345,346,…,365}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {348,349,…,364}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [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
- construction XX applied to C1 = C([345,364]), C2 = C([348,365]), C3 = C1 + C2 = C([348,364]), and C∩ = C1 ∩ C2 = C([345,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(388, 747, F3, 21) (dual of [747, 659, 22]-code), using
(67, 88, 32045)-Net in Base 3 — Upper bound on s
There is no (67, 88, 32046)-net in base 3, because
- 1 times m-reduction [i] would yield (67, 87, 32046)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 323267 703099 218689 089605 375997 766751 412781 > 387 [i]