Best Known (197, 241, s)-Nets in Base 3
(197, 241, 1480)-Net over F3 — Constructive and digital
Digital (197, 241, 1480)-net over F3, using
- 31 times duplication [i] based on digital (196, 240, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 60, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 60, 370)-net over F81, using
(197, 241, 4357)-Net over F3 — Digital
Digital (197, 241, 4357)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 4357, F3, 44) (dual of [4357, 4116, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 6594, F3, 44) (dual of [6594, 6353, 45]-code), using
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- linear OA(3233, 6561, F3, 44) (dual of [6561, 6328, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3209, 6561, F3, 40) (dual of [6561, 6352, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(43) ⊂ Ce(39) ⊂ Ce(37) [i] based on
- discarding factors / shortening the dual code based on linear OA(3241, 6594, F3, 44) (dual of [6594, 6353, 45]-code), using
(197, 241, 762881)-Net in Base 3 — Upper bound on s
There is no (197, 241, 762882)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 687788 975890 457782 188143 021106 239161 358431 093224 392173 315881 906648 425615 907800 823991 315448 221786 056410 467309 733597 > 3241 [i]