Best Known (239−27, 239, s)-Nets in Base 3
(239−27, 239, 122642)-Net over F3 — Constructive and digital
Digital (212, 239, 122642)-net over F3, using
- 33 times duplication [i] based on digital (209, 236, 122642)-net over F3, using
- net defined by OOA [i] based on linear OOA(3236, 122642, F3, 27, 27) (dual of [(122642, 27), 3311098, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3236, 1594347, F3, 27) (dual of [1594347, 1594111, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3236, 1594351, F3, 27) (dual of [1594351, 1594115, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3235, 1594324, F3, 27) (dual of [1594324, 1594089, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3236, 1594351, F3, 27) (dual of [1594351, 1594115, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3236, 1594347, F3, 27) (dual of [1594347, 1594111, 28]-code), using
- net defined by OOA [i] based on linear OOA(3236, 122642, F3, 27, 27) (dual of [(122642, 27), 3311098, 28]-NRT-code), using
(239−27, 239, 398588)-Net over F3 — Digital
Digital (212, 239, 398588)-net over F3, using
- 33 times duplication [i] based on digital (209, 236, 398588)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3236, 398588, F3, 4, 27) (dual of [(398588, 4), 1594116, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3236, 1594352, F3, 27) (dual of [1594352, 1594116, 28]-code), using
- construction X4 applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(3235, 1594324, F3, 27) (dual of [1594324, 1594089, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3209, 1594324, F3, 25) (dual of [1594324, 1594115, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324 | 326−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,13]) ⊂ C([0,12]) [i] based on
- OOA 4-folding [i] based on linear OA(3236, 1594352, F3, 27) (dual of [1594352, 1594116, 28]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3236, 398588, F3, 4, 27) (dual of [(398588, 4), 1594116, 28]-NRT-code), using
(239−27, 239, large)-Net in Base 3 — Upper bound on s
There is no (212, 239, large)-net in base 3, because
- 25 times m-reduction [i] would yield (212, 214, large)-net in base 3, but