Best Known (127, 127+13, s)-Nets in Base 5
(127, 127+13, 2796410)-Net over F5 — Constructive and digital
Digital (127, 140, 2796410)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (12, 18, 210)-net over F5, using
- net defined by OOA [i] based on linear OOA(518, 210, F5, 6, 6) (dual of [(210, 6), 1242, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- 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(513, 625, F5, 4) (dual of [625, 612, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(518, 630, F5, 6) (dual of [630, 612, 7]-code), using
- net defined by OOA [i] based on linear OOA(518, 210, F5, 6, 6) (dual of [(210, 6), 1242, 7]-NRT-code), using
- digital (109, 122, 2796200)-net over F5, using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2561, large, F25, 13) (dual of [large, large−61, 14]-code), using
- OOA 2-folding [i] based on linear OA(2561, 8388602, F25, 13) (dual of [8388602, 8388541, 14]-code), using
- trace code [i] based on linear OOA(2561, 4194301, F25, 2, 13) (dual of [(4194301, 2), 8388541, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5122, 8388602, F5, 2, 13) (dual of [(8388602, 2), 16777082, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(5122, 8388601, F5, 2, 13) (dual of [(8388601, 2), 16777080, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(5122, 2796200, F5, 14, 13) (dual of [(2796200, 14), 39146678, 14]-NRT-code), using
- digital (12, 18, 210)-net over F5, using
(127, 127+13, large)-Net over F5 — Digital
Digital (127, 140, large)-net over F5, using
- 53 times duplication [i] based on digital (124, 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
(127, 127+13, large)-Net in Base 5 — Upper bound on s
There is no (127, 140, large)-net in base 5, because
- 11 times m-reduction [i] would yield (127, 129, large)-net in base 5, but