Best Known (189, 236, s)-Nets in Base 3
(189, 236, 688)-Net over F3 — Constructive and digital
Digital (189, 236, 688)-net over F3, using
- t-expansion [i] based on digital (187, 236, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (187, 240, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 60, 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, 60, 172)-net over F81, using
- 4 times m-reduction [i] based on digital (187, 240, 688)-net over F3, using
(189, 236, 2548)-Net over F3 — Digital
Digital (189, 236, 2548)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3236, 2548, F3, 47) (dual of [2548, 2312, 48]-code), using
- 336 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0, 1, 28 times 0, 1, 34 times 0, 1, 42 times 0, 1, 47 times 0, 1, 52 times 0, 1, 55 times 0) [i] based on linear OA(3218, 2194, F3, 47) (dual of [2194, 1976, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(45) [i] based on
- linear OA(3218, 2187, F3, 47) (dual of [2187, 1969, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [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(46) ⊂ Ce(45) [i] based on
- 336 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0, 1, 15 times 0, 1, 21 times 0, 1, 28 times 0, 1, 34 times 0, 1, 42 times 0, 1, 47 times 0, 1, 52 times 0, 1, 55 times 0) [i] based on linear OA(3218, 2194, F3, 47) (dual of [2194, 1976, 48]-code), using
(189, 236, 353454)-Net in Base 3 — Upper bound on s
There is no (189, 236, 353455)-net in base 3, because
- 1 times m-reduction [i] would yield (189, 235, 353455)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13289 733560 841383 295612 652833 808051 085024 329538 819037 691532 512456 472280 445778 431871 258492 211163 160930 616781 012371 > 3235 [i]