Best Known (83−19, 83, s)-Nets in Base 3
(83−19, 83, 400)-Net over F3 — Constructive and digital
Digital (64, 83, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (64, 84, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 21, 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
- trace code for nets [i] based on digital (1, 21, 100)-net over F81, using
(83−19, 83, 703)-Net over F3 — Digital
Digital (64, 83, 703)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(383, 703, F3, 19) (dual of [703, 620, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(383, 764, F3, 19) (dual of [764, 681, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(373, 730, F3, 19) (dual of [730, 657, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(349, 730, F3, 13) (dual of [730, 681, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 312−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(310, 34, F3, 5) (dual of [34, 24, 6]-code), using
- discarding factors / shortening the dual code based on 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
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(383, 764, F3, 19) (dual of [764, 681, 20]-code), using
(83−19, 83, 46104)-Net in Base 3 — Upper bound on s
There is no (64, 83, 46105)-net in base 3, because
- 1 times m-reduction [i] would yield (64, 82, 46105)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1330 330808 752703 348082 020139 363614 456723 > 382 [i]