Best Known (105, 127, s)-Nets in Base 5
(105, 127, 7105)-Net over F5 — Constructive and digital
Digital (105, 127, 7105)-net over F5, using
- 51 times duplication [i] based on digital (104, 126, 7105)-net over F5, using
- net defined by OOA [i] based on linear OOA(5126, 7105, F5, 22, 22) (dual of [(7105, 22), 156184, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5126, 78155, F5, 22) (dual of [78155, 78029, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(5126, 78155, F5, 22) (dual of [78155, 78029, 23]-code), using
- net defined by OOA [i] based on linear OOA(5126, 7105, F5, 22, 22) (dual of [(7105, 22), 156184, 23]-NRT-code), using
(105, 127, 52558)-Net over F5 — Digital
Digital (105, 127, 52558)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5127, 52558, F5, 22) (dual of [52558, 52431, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 78155, F5, 22) (dual of [78155, 78028, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(55, 28, F5, 3) (dual of [28, 23, 4]-code or 28-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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 XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5127, 78155, F5, 22) (dual of [78155, 78028, 23]-code), using
(105, 127, large)-Net in Base 5 — Upper bound on s
There is no (105, 127, large)-net in base 5, because
- 20 times m-reduction [i] would yield (105, 107, large)-net in base 5, but