Best Known (110, 110+26, s)-Nets in Base 3
(110, 110+26, 688)-Net over F3 — Constructive and digital
Digital (110, 136, 688)-net over F3, using
- t-expansion [i] based on digital (109, 136, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 34, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 34, 172)-net over F81, using
(110, 110+26, 2309)-Net over F3 — Digital
Digital (110, 136, 2309)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 2309, F3, 26) (dual of [2309, 2173, 27]-code), using
- 99 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0) [i] based on linear OA(3120, 2194, F3, 26) (dual of [2194, 2074, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 99 step Varšamov–Edel lengthening with (ri) = (4, 1, 1, 1, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0) [i] based on linear OA(3120, 2194, F3, 26) (dual of [2194, 2074, 27]-code), using
(110, 110+26, 277829)-Net in Base 3 — Upper bound on s
There is no (110, 136, 277830)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 77355 720799 951635 200363 155290 831943 369088 418396 413168 361190 811989 > 3136 [i]