Best Known (234−47, 234, s)-Nets in Base 3
(234−47, 234, 688)-Net over F3 — Constructive and digital
Digital (187, 234, 688)-net over F3, using
- 6 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
(234−47, 234, 2437)-Net over F3 — Digital
Digital (187, 234, 2437)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3234, 2437, F3, 47) (dual of [2437, 2203, 48]-code), using
- 227 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) [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
- 227 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) [i] based on linear OA(3218, 2194, F3, 47) (dual of [2194, 1976, 48]-code), using
(234−47, 234, 321248)-Net in Base 3 — Upper bound on s
There is no (187, 234, 321249)-net in base 3, because
- 1 times m-reduction [i] would yield (187, 233, 321249)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1476 576914 216468 127570 481506 748468 203194 712957 375765 242977 892595 902406 626981 290229 568225 443018 661060 663919 274955 > 3233 [i]