Best Known (241−33, 241, s)-Nets in Base 3
(241−33, 241, 3703)-Net over F3 — Constructive and digital
Digital (208, 241, 3703)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (5, 21, 13)-net over F3, using
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 4, N(F) = 12, and 1 place with degree 2 [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (5, 12)-sequence over F3, using
- digital (187, 220, 3690)-net over F3, using
- net defined by OOA [i] based on linear OOA(3220, 3690, F3, 33, 33) (dual of [(3690, 33), 121550, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3220, 59041, F3, 33) (dual of [59041, 58821, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 59048, F3, 33) (dual of [59048, 58828, 34]-code), using
- 1 times truncation [i] based on linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 1 times truncation [i] based on linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3220, 59048, F3, 33) (dual of [59048, 58828, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3220, 59041, F3, 33) (dual of [59041, 58821, 34]-code), using
- net defined by OOA [i] based on linear OOA(3220, 3690, F3, 33, 33) (dual of [(3690, 33), 121550, 34]-NRT-code), using
- digital (5, 21, 13)-net over F3, using
(241−33, 241, 30649)-Net over F3 — Digital
Digital (208, 241, 30649)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3241, 30649, F3, 33) (dual of [30649, 30408, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 59130, F3, 33) (dual of [59130, 58889, 34]-code), using
- 4 times code embedding in larger space [i] based on linear OA(3237, 59126, F3, 33) (dual of [59126, 58889, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- linear OA(3221, 59050, F3, 33) (dual of [59050, 58829, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3161, 59050, F3, 25) (dual of [59050, 58889, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(316, 76, F3, 7) (dual of [76, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to C([0,16]) ⊂ C([0,12]) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(3237, 59126, F3, 33) (dual of [59126, 58889, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3241, 59130, F3, 33) (dual of [59130, 58889, 34]-code), using
(241−33, 241, large)-Net in Base 3 — Upper bound on s
There is no (208, 241, large)-net in base 3, because
- 31 times m-reduction [i] would yield (208, 210, large)-net in base 3, but