Best Known (108, 108+36, s)-Nets in Base 3
(108, 108+36, 328)-Net over F3 — Constructive and digital
Digital (108, 144, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 36, 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
(108, 108+36, 658)-Net over F3 — Digital
Digital (108, 144, 658)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3144, 658, F3, 36) (dual of [658, 514, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 748, F3, 36) (dual of [748, 604, 37]-code), using
- construction XX applied to C1 = C([725,31]), C2 = C([0,33]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([725,33]) [i] based on
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,31}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,33], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3142, 728, F3, 37) (dual of [728, 586, 38]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,33}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- 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(31, 7, F3, 1) (dual of [7, 6, 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 (see above)
- construction XX applied to C1 = C([725,31]), C2 = C([0,33]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([725,33]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3144, 748, F3, 36) (dual of [748, 604, 37]-code), using
(108, 108+36, 24760)-Net in Base 3 — Upper bound on s
There is no (108, 144, 24761)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 507 630245 371446 188821 860612 124455 719733 232530 633393 941658 920688 346617 > 3144 [i]