Best Known (12, 12+11, s)-Nets in Base 256
(12, 12+11, 13108)-Net over F256 — Constructive and digital
Digital (12, 23, 13108)-net over F256, using
- 2561 times duplication [i] based on digital (11, 22, 13108)-net over F256, using
- net defined by OOA [i] based on linear OOA(25622, 13108, F256, 11, 11) (dual of [(13108, 11), 144166, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25622, 65541, F256, 11) (dual of [65541, 65519, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(25617, 65537, F256, 9) (dual of [65537, 65520, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25622, 65542, F256, 11) (dual of [65542, 65520, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25622, 65541, F256, 11) (dual of [65541, 65519, 12]-code), using
- net defined by OOA [i] based on linear OOA(25622, 13108, F256, 11, 11) (dual of [(13108, 11), 144166, 12]-NRT-code), using
(12, 12+11, 30955)-Net over F256 — Digital
Digital (12, 23, 30955)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25623, 30955, F256, 2, 11) (dual of [(30955, 2), 61887, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25623, 32772, F256, 2, 11) (dual of [(32772, 2), 65521, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25623, 65544, F256, 11) (dual of [65544, 65521, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(10) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(25623, 65544, F256, 11) (dual of [65544, 65521, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(25623, 32772, F256, 2, 11) (dual of [(32772, 2), 65521, 12]-NRT-code), using
(12, 12+11, large)-Net in Base 256 — Upper bound on s
There is no (12, 23, large)-net in base 256, because
- 9 times m-reduction [i] would yield (12, 14, large)-net in base 256, but