Best Known (27−13, 27, s)-Nets in Base 256
(27−13, 27, 10923)-Net over F256 — Constructive and digital
Digital (14, 27, 10923)-net over F256, using
- 2561 times duplication [i] based on digital (13, 26, 10923)-net over F256, using
- net defined by OOA [i] based on linear OOA(25626, 10923, F256, 13, 13) (dual of [(10923, 13), 141973, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25626, 65539, F256, 13) (dual of [65539, 65513, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- linear OA(25625, 65537, F256, 13) (dual of [65537, 65512, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25626, 65542, F256, 13) (dual of [65542, 65516, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(25626, 65539, F256, 13) (dual of [65539, 65513, 14]-code), using
- net defined by OOA [i] based on linear OOA(25626, 10923, F256, 13, 13) (dual of [(10923, 13), 141973, 14]-NRT-code), using
(27−13, 27, 21848)-Net over F256 — Digital
Digital (14, 27, 21848)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25627, 21848, F256, 3, 13) (dual of [(21848, 3), 65517, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25627, 65544, F256, 13) (dual of [65544, 65517, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(25625, 65536, F256, 13) (dual of [65536, 65511, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(25627, 65544, F256, 13) (dual of [65544, 65517, 14]-code), using
(27−13, 27, large)-Net in Base 256 — Upper bound on s
There is no (14, 27, large)-net in base 256, because
- 11 times m-reduction [i] would yield (14, 16, large)-net in base 256, but