Best Known (174, 198, s)-Nets in Base 3
(174, 198, 44289)-Net over F3 — Constructive and digital
Digital (174, 198, 44289)-net over F3, using
- 31 times duplication [i] based on digital (173, 197, 44289)-net over F3, using
- t-expansion [i] based on digital (172, 197, 44289)-net over F3, using
- net defined by OOA [i] based on linear OOA(3197, 44289, F3, 25, 25) (dual of [(44289, 25), 1107028, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3197, 531469, F3, 25) (dual of [531469, 531272, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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(24) ⊂ Ce(21) [i] based on
- OOA 12-folding and stacking with additional row [i] based on linear OA(3197, 531469, F3, 25) (dual of [531469, 531272, 26]-code), using
- net defined by OOA [i] based on linear OOA(3197, 44289, F3, 25, 25) (dual of [(44289, 25), 1107028, 26]-NRT-code), using
- t-expansion [i] based on digital (172, 197, 44289)-net over F3, using
(174, 198, 177157)-Net over F3 — Digital
Digital (174, 198, 177157)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3198, 177157, F3, 3, 24) (dual of [(177157, 3), 531273, 25]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3195, 177156, F3, 3, 24) (dual of [(177156, 3), 531273, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3195, 531468, F3, 24) (dual of [531468, 531273, 25]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3194, 531467, F3, 24) (dual of [531467, 531273, 25]-code), using
- construction X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(3193, 531441, F3, 25) (dual of [531441, 531248, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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 X4 applied to Ce(24) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3194, 531467, F3, 24) (dual of [531467, 531273, 25]-code), using
- OOA 3-folding [i] based on linear OA(3195, 531468, F3, 24) (dual of [531468, 531273, 25]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3195, 177156, F3, 3, 24) (dual of [(177156, 3), 531273, 25]-NRT-code), using
(174, 198, large)-Net in Base 3 — Upper bound on s
There is no (174, 198, large)-net in base 3, because
- 22 times m-reduction [i] would yield (174, 176, large)-net in base 3, but