Best Known (51, 51+11, s)-Nets in Base 25
(51, 51+11, 1678621)-Net over F25 — Constructive and digital
Digital (51, 62, 1678621)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (6, 11, 901)-net over F25, using
- net defined by OOA [i] based on linear OOA(2511, 901, F25, 5, 5) (dual of [(901, 5), 4494, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(2511, 1803, F25, 5) (dual of [1803, 1792, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(251, 601, F25, 1) (dual of [601, 600, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(253, 601, F25, 2) (dual of [601, 598, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(253, 624, F25, 2) (dual of [624, 621, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(253, 624, F25, 2) (dual of [624, 621, 3]-code), using
- linear OA(257, 601, F25, 5) (dual of [601, 594, 6]-code), using
- linear OA(251, 601, F25, 1) (dual of [601, 600, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(2511, 1803, F25, 5) (dual of [1803, 1792, 6]-code), using
- net defined by OOA [i] based on linear OOA(2511, 901, F25, 5, 5) (dual of [(901, 5), 4494, 6]-NRT-code), using
- digital (40, 51, 1677720)-net over F25, using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2551, large, F25, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2551, 8388601, F25, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2551, 1677720, F25, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- digital (6, 11, 901)-net over F25, using
(51, 51+11, large)-Net over F25 — Digital
Digital (51, 62, large)-net over F25, using
- 251 times duplication [i] based on digital (50, 61, large)-net over F25, using
- t-expansion [i] based on digital (48, 61, large)-net over F25, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 2510−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- t-expansion [i] based on digital (48, 61, large)-net over F25, using
(51, 51+11, large)-Net in Base 25 — Upper bound on s
There is no (51, 62, large)-net in base 25, because
- 9 times m-reduction [i] would yield (51, 53, large)-net in base 25, but