Best Known (144−15, 144, s)-Nets in Base 3
(144−15, 144, 683285)-Net over F3 — Constructive and digital
Digital (129, 144, 683285)-net over F3, using
- 32 times duplication [i] based on digital (127, 142, 683285)-net over F3, using
- net defined by OOA [i] based on linear OOA(3142, 683285, F3, 15, 15) (dual of [(683285, 15), 10249133, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3142, 4782996, F3, 15) (dual of [4782996, 4782854, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3142, 4782999, F3, 15) (dual of [4782999, 4782857, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 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(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(31, 29, F3, 1) (dual of [29, 28, 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 X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3142, 4782999, F3, 15) (dual of [4782999, 4782857, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3142, 4782996, F3, 15) (dual of [4782996, 4782854, 16]-code), using
- net defined by OOA [i] based on linear OOA(3142, 683285, F3, 15, 15) (dual of [(683285, 15), 10249133, 16]-NRT-code), using
(144−15, 144, 1594334)-Net over F3 — Digital
Digital (129, 144, 1594334)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3144, 1594334, F3, 3, 15) (dual of [(1594334, 3), 4782858, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3144, 4783002, F3, 15) (dual of [4783002, 4782858, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3142, 4783000, F3, 15) (dual of [4783000, 4782858, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- 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(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(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([0,7]) ⊂ C([0,6]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3142, 4783000, F3, 15) (dual of [4783000, 4782858, 16]-code), using
- OOA 3-folding [i] based on linear OA(3144, 4783002, F3, 15) (dual of [4783002, 4782858, 16]-code), using
(144−15, 144, large)-Net in Base 3 — Upper bound on s
There is no (129, 144, large)-net in base 3, because
- 13 times m-reduction [i] would yield (129, 131, large)-net in base 3, but