Best Known (125, 125+11, s)-Nets in Base 5
(125, 125+11, 5033160)-Net over F5 — Constructive and digital
Digital (125, 136, 5033160)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (12, 15, 1677720)-net over F5, using
- s-reduction based on digital (12, 15, 3050326)-net over F5, using
- net defined by OOA [i] based on linear OOA(515, 3050326, F5, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(515, 3050326, F5, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
- OAs with strength 3, b ≠ 2, and m > 3 are always embeddable [i] based on linear OA(515, 3050326, F5, 3) (dual of [3050326, 3050311, 4]-code or 3050326-cap in PG(14,5)), using
- appending kth column [i] based on linear OOA(515, 3050326, F5, 2, 3) (dual of [(3050326, 2), 6100637, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(515, 3050326, F5, 3, 3) (dual of [(3050326, 3), 9150963, 4]-NRT-code), using
- s-reduction based on digital (12, 15, 3050326)-net over F5, using
- digital (35, 40, 1677720)-net over F5, using
- s-reduction based on digital (35, 40, 4194301)-net over F5, using
- net defined by OOA [i] based on linear OOA(540, 4194301, F5, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(540, large, F5, 5) (dual of [large, large−40, 6]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(540, large, F5, 5) (dual of [large, large−40, 6]-code), using
- net defined by OOA [i] based on linear OOA(540, 4194301, F5, 5, 5) (dual of [(4194301, 5), 20971465, 6]-NRT-code), using
- s-reduction based on digital (35, 40, 4194301)-net over F5, using
- digital (70, 81, 1677720)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, 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(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- digital (12, 15, 1677720)-net over F5, using
(125, 125+11, large)-Net over F5 — Digital
Digital (125, 136, large)-net over F5, using
- t-expansion [i] based on digital (121, 136, large)-net over F5, using
- 1 times m-reduction [i] based on digital (121, 137, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5137, large, F5, 16) (dual of [large, large−137, 17]-code), using
- 16 times code embedding in larger space [i] based on linear OA(5121, large, F5, 16) (dual of [large, large−121, 17]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 16 times code embedding in larger space [i] based on linear OA(5121, large, F5, 16) (dual of [large, large−121, 17]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(5137, large, F5, 16) (dual of [large, large−137, 17]-code), using
- 1 times m-reduction [i] based on digital (121, 137, large)-net over F5, using
(125, 125+11, large)-Net in Base 5 — Upper bound on s
There is no (125, 136, large)-net in base 5, because
- 9 times m-reduction [i] would yield (125, 127, large)-net in base 5, but