Best Known (123, 123+15, s)-Nets in Base 5
(123, 123+15, 1198471)-Net over F5 — Constructive and digital
Digital (123, 138, 1198471)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (11, 18, 100)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 25)-net over F5, using
- s-reduction based on digital (0, 1, s)-net over F5 with arbitrarily large s, using
- digital (1, 3, 25)-net over F5, using
- s-reduction based on digital (1, 3, 31)-net over F5, using
- digital (1, 4, 25)-net over F5, using
- s-reduction based on digital (1, 4, 26)-net over F5, using
- net defined by OOA [i] based on linear OOA(54, 26, F5, 3, 3) (dual of [(26, 3), 74, 4]-NRT-code), using
- s-reduction based on digital (1, 4, 26)-net over F5, using
- digital (3, 10, 25)-net over F5, using
- digital (0, 1, 25)-net over F5, using
- generalized (u, u+v)-construction [i] based on
- digital (105, 120, 1198371)-net over F5, using
- net defined by OOA [i] based on linear OOA(5120, 1198371, F5, 15, 15) (dual of [(1198371, 15), 17975445, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5120, 8388598, F5, 15) (dual of [8388598, 8388478, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(5120, large, F5, 15) (dual of [large, large−120, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5120, 8388598, F5, 15) (dual of [8388598, 8388478, 16]-code), using
- net defined by OOA [i] based on linear OOA(5120, 1198371, F5, 15, 15) (dual of [(1198371, 15), 17975445, 16]-NRT-code), using
- digital (11, 18, 100)-net over F5, using
(123, 123+15, large)-Net over F5 — Digital
Digital (123, 138, large)-net over F5, using
- 51 times duplication [i] based on digital (122, 137, large)-net over F5, using
- t-expansion [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
- t-expansion [i] based on digital (121, 137, large)-net over F5, using
(123, 123+15, large)-Net in Base 5 — Upper bound on s
There is no (123, 138, large)-net in base 5, because
- 13 times m-reduction [i] would yield (123, 125, large)-net in base 5, but