Best Known (135−15, 135, s)-Nets in Base 3
(135−15, 135, 227764)-Net over F3 — Constructive and digital
Digital (120, 135, 227764)-net over F3, using
- 33 times duplication [i] based on digital (117, 132, 227764)-net over F3, using
- net defined by OOA [i] based on linear OOA(3132, 227764, F3, 15, 15) (dual of [(227764, 15), 3416328, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3132, 1594349, F3, 15) (dual of [1594349, 1594217, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(3132, 1594351, F3, 15) (dual of [1594351, 1594219, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3131, 1594324, F3, 15) (dual of [1594324, 1594193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3105, 1594324, F3, 13) (dual of [1594324, 1594219, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(31, 27, F3, 1) (dual of [27, 26, 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(3132, 1594351, F3, 15) (dual of [1594351, 1594219, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(3132, 1594349, F3, 15) (dual of [1594349, 1594217, 16]-code), using
- net defined by OOA [i] based on linear OOA(3132, 227764, F3, 15, 15) (dual of [(227764, 15), 3416328, 16]-NRT-code), using
(135−15, 135, 531451)-Net over F3 — Digital
Digital (120, 135, 531451)-net over F3, using
- 32 times duplication [i] based on digital (118, 133, 531451)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 531451, F3, 3, 15) (dual of [(531451, 3), 1594220, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3133, 1594353, F3, 15) (dual of [1594353, 1594220, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3132, 1594352, F3, 15) (dual of [1594352, 1594220, 16]-code), using
- construction X4 applied to C([0,7]) ⊂ C([0,6]) [i] based on
- linear OA(3131, 1594324, F3, 15) (dual of [1594324, 1594193, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(3105, 1594324, F3, 13) (dual of [1594324, 1594219, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(327, 28, F3, 27) (dual of [28, 1, 28]-code or 28-arc in PG(26,3)), using
- dual of repetition code with length 28 [i]
- linear OA(31, 28, F3, 1) (dual of [28, 27, 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
- 1 times code embedding in larger space [i] based on linear OA(3132, 1594352, F3, 15) (dual of [1594352, 1594220, 16]-code), using
- OOA 3-folding [i] based on linear OA(3133, 1594353, F3, 15) (dual of [1594353, 1594220, 16]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 531451, F3, 3, 15) (dual of [(531451, 3), 1594220, 16]-NRT-code), using
(135−15, 135, large)-Net in Base 3 — Upper bound on s
There is no (120, 135, large)-net in base 3, because
- 13 times m-reduction [i] would yield (120, 122, large)-net in base 3, but