Best Known (157, 176, s)-Nets in Base 3
(157, 176, 531444)-Net over F3 — Constructive and digital
Digital (157, 176, 531444)-net over F3, using
- 33 times duplication [i] based on digital (154, 173, 531444)-net over F3, using
- net defined by OOA [i] based on linear OOA(3173, 531444, F3, 19, 19) (dual of [(531444, 19), 10097263, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3173, 4782997, F3, 19) (dual of [4782997, 4782824, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3173, 4783001, F3, 19) (dual of [4783001, 4782828, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3169, 4782969, F3, 19) (dual of [4782969, 4782800, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3141, 4782969, F3, 16) (dual of [4782969, 4782828, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3173, 4783001, F3, 19) (dual of [4783001, 4782828, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3173, 4782997, F3, 19) (dual of [4782997, 4782824, 20]-code), using
- net defined by OOA [i] based on linear OOA(3173, 531444, F3, 19, 19) (dual of [(531444, 19), 10097263, 20]-NRT-code), using
(157, 176, 1195751)-Net over F3 — Digital
Digital (157, 176, 1195751)-net over F3, using
- 31 times duplication [i] based on digital (156, 175, 1195751)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3175, 1195751, F3, 4, 19) (dual of [(1195751, 4), 4782829, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3175, 4783004, F3, 19) (dual of [4783004, 4782829, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- 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(3141, 4782970, F3, 15) (dual of [4782970, 4782829, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(36, 34, F3, 3) (dual of [34, 28, 4]-code or 34-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([0,9]) ⊂ C([0,7]) [i] based on
- OOA 4-folding [i] based on linear OA(3175, 4783004, F3, 19) (dual of [4783004, 4782829, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3175, 1195751, F3, 4, 19) (dual of [(1195751, 4), 4782829, 20]-NRT-code), using
(157, 176, large)-Net in Base 3 — Upper bound on s
There is no (157, 176, large)-net in base 3, because
- 17 times m-reduction [i] would yield (157, 159, large)-net in base 3, but