Best Known (207−31, 207, s)-Nets in Base 3
(207−31, 207, 3938)-Net over F3 — Constructive and digital
Digital (176, 207, 3938)-net over F3, using
- 32 times duplication [i] based on digital (174, 205, 3938)-net over F3, using
- net defined by OOA [i] based on linear OOA(3205, 3938, F3, 31, 31) (dual of [(3938, 31), 121873, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3205, 59071, F3, 31) (dual of [59071, 58866, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3205, 59073, F3, 31) (dual of [59073, 58868, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3205, 59073, F3, 31) (dual of [59073, 58868, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3205, 59071, F3, 31) (dual of [59071, 58866, 32]-code), using
- net defined by OOA [i] based on linear OOA(3205, 3938, F3, 31, 31) (dual of [(3938, 31), 121873, 32]-NRT-code), using
(207−31, 207, 19692)-Net over F3 — Digital
Digital (176, 207, 19692)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3207, 19692, F3, 3, 31) (dual of [(19692, 3), 58869, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3207, 59076, F3, 31) (dual of [59076, 58869, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3201, 59050, F3, 31) (dual of [59050, 58849, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3181, 59050, F3, 27) (dual of [59050, 58869, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (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([0,15]) ⊂ C([0,13]) [i] based on
- OOA 3-folding [i] based on linear OA(3207, 59076, F3, 31) (dual of [59076, 58869, 32]-code), using
(207−31, 207, large)-Net in Base 3 — Upper bound on s
There is no (176, 207, large)-net in base 3, because
- 29 times m-reduction [i] would yield (176, 178, large)-net in base 3, but