Best Known (197, 249, s)-Nets in Base 3
(197, 249, 688)-Net over F3 — Constructive and digital
Digital (197, 249, 688)-net over F3, using
- 31 times duplication [i] based on digital (196, 248, 688)-net over F3, using
- t-expansion [i] based on digital (193, 248, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 62, 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, 62, 172)-net over F81, using
- t-expansion [i] based on digital (193, 248, 688)-net over F3, using
(197, 249, 2219)-Net over F3 — Digital
Digital (197, 249, 2219)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 2219, F3, 52) (dual of [2219, 1970, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 2223, F3, 52) (dual of [2223, 1974, 53]-code), using
- construction X applied to Ce(51) ⊂ Ce(45) [i] based on
- linear OA(3239, 2187, F3, 52) (dual of [2187, 1948, 53]-code), using an extension Ce(51) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,51], and designed minimum distance d ≥ |I|+1 = 52 [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(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 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(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- construction X applied to Ce(51) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(3249, 2223, F3, 52) (dual of [2223, 1974, 53]-code), using
(197, 249, 195686)-Net in Base 3 — Upper bound on s
There is no (197, 249, 195687)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 63568 718943 040761 461344 931517 083483 907779 491400 439445 444376 050294 761907 352949 453014 420518 604904 520414 115103 682878 685725 > 3249 [i]