Best Known (176, 176+30, s)-Nets in Base 3
(176, 176+30, 3938)-Net over F3 — Constructive and digital
Digital (176, 206, 3938)-net over F3, using
- 31 times duplication [i] based on digital (175, 205, 3938)-net over F3, using
- t-expansion [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
- t-expansion [i] based on digital (174, 205, 3938)-net over F3, using
(176, 176+30, 21967)-Net over F3 — Digital
Digital (176, 206, 21967)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3206, 21967, F3, 2, 30) (dual of [(21967, 2), 43728, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 29538, F3, 2, 30) (dual of [(29538, 2), 58870, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3206, 59076, F3, 30) (dual of [59076, 58870, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3206, 59077, F3, 30) (dual of [59077, 58871, 31]-code), using
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [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(3171, 59049, F3, 26) (dual of [59049, 58878, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(30) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3206, 59077, F3, 30) (dual of [59077, 58871, 31]-code), using
- OOA 2-folding [i] based on linear OA(3206, 59076, F3, 30) (dual of [59076, 58870, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 29538, F3, 2, 30) (dual of [(29538, 2), 58870, 31]-NRT-code), using
(176, 176+30, large)-Net in Base 3 — Upper bound on s
There is no (176, 206, large)-net in base 3, because
- 28 times m-reduction [i] would yield (176, 178, large)-net in base 3, but