Best Known (76, 76+19, s)-Nets in Base 3
(76, 76+19, 600)-Net over F3 — Constructive and digital
Digital (76, 95, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (76, 96, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 24, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 24, 150)-net over F81, using
(76, 76+19, 1544)-Net over F3 — Digital
Digital (76, 95, 1544)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(395, 1544, F3, 19) (dual of [1544, 1449, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(395, 2224, F3, 19) (dual of [2224, 2129, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(357, 2188, F3, 13) (dual of [2188, 2131, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(395, 2224, F3, 19) (dual of [2224, 2129, 20]-code), using
(76, 76+19, 199512)-Net in Base 3 — Upper bound on s
There is no (76, 95, 199513)-net in base 3, because
- 1 times m-reduction [i] would yield (76, 94, 199513)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 706 991570 357806 529101 971615 708914 830253 223443 > 394 [i]