Best Known (115−25, 115, s)-Nets in Base 3
(115−25, 115, 464)-Net over F3 — Constructive and digital
Digital (90, 115, 464)-net over F3, using
- t-expansion [i] based on digital (89, 115, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (89, 116, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 29, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 29, 116)-net over F81, using
- 1 times m-reduction [i] based on digital (89, 116, 464)-net over F3, using
(115−25, 115, 1098)-Net over F3 — Digital
Digital (90, 115, 1098)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3115, 1098, F3, 2, 25) (dual of [(1098, 2), 2081, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 2196, F3, 25) (dual of [2196, 2081, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3115, 2197, F3, 25) (dual of [2197, 2082, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3115, 2197, F3, 25) (dual of [2197, 2082, 26]-code), using
- OOA 2-folding [i] based on linear OA(3115, 2196, F3, 25) (dual of [2196, 2081, 26]-code), using
(115−25, 115, 90142)-Net in Base 3 — Upper bound on s
There is no (90, 115, 90143)-net in base 3, because
- 1 times m-reduction [i] would yield (90, 114, 90143)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 465311 818316 075766 516345 268208 239627 979019 251437 609449 > 3114 [i]