Best Known (126, 126+15, s)-Nets in Base 5
(126, 126+15, 1198580)-Net over F5 — Constructive and digital
Digital (126, 141, 1198580)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (14, 21, 209)-net over F5, using
- net defined by OOA [i] based on linear OOA(521, 209, F5, 7, 7) (dual of [(209, 7), 1442, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(521, 625, F5, 7) (dual of [625, 604, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 625, F5, 6) (dual of [625, 608, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
- net defined by OOA [i] based on linear OOA(521, 209, F5, 7, 7) (dual of [(209, 7), 1442, 8]-NRT-code), using
- 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 (14, 21, 209)-net over F5, using
(126, 126+15, large)-Net over F5 — Digital
Digital (126, 141, large)-net over F5, using
- 54 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
(126, 126+15, large)-Net in Base 5 — Upper bound on s
There is no (126, 141, large)-net in base 5, because
- 13 times m-reduction [i] would yield (126, 128, large)-net in base 5, but