Best Known (210, 210+37, s)-Nets in Base 3
(210, 210+37, 3281)-Net over F3 — Constructive and digital
Digital (210, 247, 3281)-net over F3, using
- 35 times duplication [i] based on digital (205, 242, 3281)-net over F3, using
- net defined by OOA [i] based on linear OOA(3242, 3281, F3, 37, 37) (dual of [(3281, 37), 121155, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3242, 59059, F3, 37) (dual of [59059, 58817, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3242, 59060, F3, 37) (dual of [59060, 58818, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(3241, 59049, F3, 37) (dual of [59049, 58808, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3231, 59049, F3, 35) (dual of [59049, 58818, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3242, 59060, F3, 37) (dual of [59060, 58818, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(3242, 59059, F3, 37) (dual of [59059, 58817, 38]-code), using
- net defined by OOA [i] based on linear OOA(3242, 3281, F3, 37, 37) (dual of [(3281, 37), 121155, 38]-NRT-code), using
(210, 210+37, 19692)-Net over F3 — Digital
Digital (210, 247, 19692)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3247, 19692, F3, 3, 37) (dual of [(19692, 3), 58829, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(3241, 59050, F3, 37) (dual of [59050, 58809, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(36, 26, F3, 3) (dual of [26, 20, 4]-code or 26-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
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- OOA 3-folding [i] based on linear OA(3247, 59076, F3, 37) (dual of [59076, 58829, 38]-code), using
(210, 210+37, large)-Net in Base 3 — Upper bound on s
There is no (210, 247, large)-net in base 3, because
- 35 times m-reduction [i] would yield (210, 212, large)-net in base 3, but