Best Known (113, 113+29, s)-Nets in Base 3
(113, 113+29, 640)-Net over F3 — Constructive and digital
Digital (113, 142, 640)-net over F3, using
- 2 times m-reduction [i] based on digital (113, 144, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 36, 160)-net over F81, using
(113, 113+29, 1669)-Net over F3 — Digital
Digital (113, 142, 1669)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3142, 1669, F3, 29) (dual of [1669, 1527, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 2203, F3, 29) (dual of [2203, 2061, 30]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3141, 2188, F3, 31) (dual of [2188, 2047, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3127, 2188, F3, 27) (dual of [2188, 2061, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(31, 15, F3, 1) (dual of [15, 14, 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
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3142, 2203, F3, 29) (dual of [2203, 2061, 30]-code), using
(113, 113+29, 193058)-Net in Base 3 — Upper bound on s
There is no (113, 142, 193059)-net in base 3, because
- 1 times m-reduction [i] would yield (113, 141, 193059)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 18 797385 839098 520235 073828 824559 118734 299498 926844 236568 857224 511373 > 3141 [i]