Best Known (248−52, 248, s)-Nets in Base 3
(248−52, 248, 688)-Net over F3 — Constructive and digital
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
(248−52, 248, 2170)-Net over F3 — Digital
Digital (196, 248, 2170)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3248, 2170, F3, 52) (dual of [2170, 1922, 53]-code), using
- discarding factors / shortening the dual code based on linear OA(3248, 2218, F3, 52) (dual of [2218, 1970, 53]-code), using
- construction XX applied to Ce(51) ⊂ Ce(46) ⊂ 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(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(38, 30, F3, 4) (dual of [30, 22, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(51) ⊂ Ce(46) ⊂ Ce(45) [i] based on
- discarding factors / shortening the dual code based on linear OA(3248, 2218, F3, 52) (dual of [2218, 1970, 53]-code), using
(248−52, 248, 187588)-Net in Base 3 — Upper bound on s
There is no (196, 248, 187589)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 21187 864193 821739 908598 180928 345236 046580 322887 334415 423977 644101 848105 374565 339052 608894 418511 387917 720707 954350 358961 > 3248 [i]