Best Known (216−45, 216, s)-Nets in Base 3
(216−45, 216, 688)-Net over F3 — Constructive and digital
Digital (171, 216, 688)-net over F3, using
- t-expansion [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(216−45, 216, 2012)-Net over F3 — Digital
Digital (171, 216, 2012)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 2012, F3, 45) (dual of [2012, 1796, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 2209, F3, 45) (dual of [2209, 1993, 46]-code), using
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- 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(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(31, 18, F3, 1) (dual of [18, 17, 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(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(45) ⊂ Ce(42) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 2209, F3, 45) (dual of [2209, 1993, 46]-code), using
(216−45, 216, 208235)-Net in Base 3 — Upper bound on s
There is no (171, 216, 208236)-net in base 3, because
- 1 times m-reduction [i] would yield (171, 215, 208236)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 3 811602 331994 723810 900940 453040 529924 621327 557281 699903 062634 861147 578740 356852 820536 229426 259520 862281 > 3215 [i]