Best Known (173−30, 173, s)-Nets in Base 3
(173−30, 173, 896)-Net over F3 — Constructive and digital
Digital (143, 173, 896)-net over F3, using
- 31 times duplication [i] based on digital (142, 172, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 43, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 43, 224)-net over F81, using
(173−30, 173, 4791)-Net over F3 — Digital
Digital (143, 173, 4791)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3173, 4791, F3, 30) (dual of [4791, 4618, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3173, 6606, F3, 30) (dual of [6606, 6433, 31]-code), using
- strength reduction [i] based on linear OA(3173, 6606, F3, 31) (dual of [6606, 6433, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(312, 44, F3, 5) (dual of [44, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- strength reduction [i] based on linear OA(3173, 6606, F3, 31) (dual of [6606, 6433, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3173, 6606, F3, 30) (dual of [6606, 6433, 31]-code), using
(173−30, 173, 1022183)-Net in Base 3 — Upper bound on s
There is no (143, 173, 1022184)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 34832 040219 294961 161547 678549 944448 707966 292036 092684 453245 738487 062334 791715 823841 > 3173 [i]