Best Known (178, 178+21, s)-Nets in Base 3
(178, 178+21, 478300)-Net over F3 — Constructive and digital
Digital (178, 199, 478300)-net over F3, using
- net defined by OOA [i] based on linear OOA(3199, 478300, F3, 21, 21) (dual of [(478300, 21), 10044101, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3199, 4783001, F3, 21) (dual of [4783001, 4782802, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3198, 4783000, F3, 21) (dual of [4783000, 4782802, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3197, 4782970, F3, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3198, 4783000, F3, 21) (dual of [4783000, 4782802, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3199, 4783001, F3, 21) (dual of [4783001, 4782802, 22]-code), using
(178, 178+21, 1195750)-Net over F3 — Digital
Digital (178, 199, 1195750)-net over F3, using
- 31 times duplication [i] based on digital (177, 198, 1195750)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3198, 1195750, F3, 4, 21) (dual of [(1195750, 4), 4782802, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3198, 4783000, F3, 21) (dual of [4783000, 4782802, 22]-code), using
- construction X4 applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3197, 4782970, F3, 21) (dual of [4782970, 4782773, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(3169, 4782970, F3, 19) (dual of [4782970, 4782801, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(329, 30, F3, 29) (dual of [30, 1, 30]-code or 30-arc in PG(28,3)), using
- dual of repetition code with length 30 [i]
- linear OA(31, 30, F3, 1) (dual of [30, 29, 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 C([0,10]) ⊂ C([0,9]) [i] based on
- OOA 4-folding [i] based on linear OA(3198, 4783000, F3, 21) (dual of [4783000, 4782802, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3198, 1195750, F3, 4, 21) (dual of [(1195750, 4), 4782802, 22]-NRT-code), using
(178, 178+21, large)-Net in Base 3 — Upper bound on s
There is no (178, 199, large)-net in base 3, because
- 19 times m-reduction [i] would yield (178, 180, large)-net in base 3, but