Best Known (112, 112+31, s)-Nets in Base 3
(112, 112+31, 600)-Net over F3 — Constructive and digital
Digital (112, 143, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (112, 144, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 36, 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, 36, 150)-net over F81, using
(112, 112+31, 1239)-Net over F3 — Digital
Digital (112, 143, 1239)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3143, 1239, F3, 31) (dual of [1239, 1096, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3143, 2197, F3, 31) (dual of [2197, 2054, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(31, 9, F3, 1) (dual of [9, 8, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3143, 2197, F3, 31) (dual of [2197, 2054, 32]-code), using
(112, 112+31, 105541)-Net in Base 3 — Upper bound on s
There is no (112, 143, 105542)-net in base 3, because
- 1 times m-reduction [i] would yield (112, 142, 105542)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 56 392973 753933 568780 689489 986924 798717 987706 573000 049211 915447 189705 > 3142 [i]