Best Known (50−25, 50, s)-Nets in Base 256
(50−25, 50, 5461)-Net over F256 — Constructive and digital
Digital (25, 50, 5461)-net over F256, using
- 2561 times duplication [i] based on digital (24, 49, 5461)-net over F256, using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(25649, 65536, F256, 25) (dual of [65536, 65487, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(25649, 65533, F256, 25) (dual of [65533, 65484, 26]-code), using
- net defined by OOA [i] based on linear OOA(25649, 5461, F256, 25, 25) (dual of [(5461, 25), 136476, 26]-NRT-code), using
(50−25, 50, 13108)-Net over F256 — Digital
Digital (25, 50, 13108)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25650, 13108, F256, 5, 25) (dual of [(13108, 5), 65490, 26]-NRT-code), using
- OOA 5-folding [i] based on linear OA(25650, 65540, F256, 25) (dual of [65540, 65490, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(25650, 65542, F256, 25) (dual of [65542, 65492, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(25650, 65542, F256, 25) (dual of [65542, 65492, 26]-code), using
- OOA 5-folding [i] based on linear OA(25650, 65540, F256, 25) (dual of [65540, 65490, 26]-code), using
(50−25, 50, large)-Net in Base 256 — Upper bound on s
There is no (25, 50, large)-net in base 256, because
- 23 times m-reduction [i] would yield (25, 27, large)-net in base 256, but