Best Known (117−12, 117, s)-Nets in Base 3
(117−12, 117, 797167)-Net over F3 — Constructive and digital
Digital (105, 117, 797167)-net over F3, using
- net defined by OOA [i] based on linear OOA(3117, 797167, F3, 12, 12) (dual of [(797167, 12), 9565887, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3117, 4783002, F3, 12) (dual of [4783002, 4782885, 13]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- linear OA(3113, 4782970, F3, 13) (dual of [4782970, 4782857, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(385, 4782970, F3, 9) (dual of [4782970, 4782885, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4782970 | 328−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [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 C([0,6]) ⊂ C([0,4]) [i] based on
- OA 6-folding and stacking [i] based on linear OA(3117, 4783002, F3, 12) (dual of [4783002, 4782885, 13]-code), using
(117−12, 117, 2391501)-Net over F3 — Digital
Digital (105, 117, 2391501)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3117, 2391501, F3, 2, 12) (dual of [(2391501, 2), 4782885, 13]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3115, 2391500, F3, 2, 12) (dual of [(2391500, 2), 4782885, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3115, 4783000, F3, 12) (dual of [4783000, 4782885, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3114, 4782999, F3, 12) (dual of [4782999, 4782885, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(3113, 4782969, F3, 13) (dual of [4782969, 4782856, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(385, 4782969, F3, 10) (dual of [4782969, 4782884, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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 Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3114, 4782999, F3, 12) (dual of [4782999, 4782885, 13]-code), using
- OOA 2-folding [i] based on linear OA(3115, 4783000, F3, 12) (dual of [4783000, 4782885, 13]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3115, 2391500, F3, 2, 12) (dual of [(2391500, 2), 4782885, 13]-NRT-code), using
(117−12, 117, large)-Net in Base 3 — Upper bound on s
There is no (105, 117, large)-net in base 3, because
- 10 times m-reduction [i] would yield (105, 107, large)-net in base 3, but