Best Known (104, 104+23, s)-Nets in Base 5
(104, 104+23, 7102)-Net over F5 — Constructive and digital
Digital (104, 127, 7102)-net over F5, using
- net defined by OOA [i] based on linear OOA(5127, 7102, F5, 23, 23) (dual of [(7102, 23), 163219, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5127, 78123, F5, 23) (dual of [78123, 77996, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5127, 78123, F5, 23) (dual of [78123, 77996, 24]-code), using
(104, 104+23, 39066)-Net over F5 — Digital
Digital (104, 127, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5127, 39066, F5, 2, 23) (dual of [(39066, 2), 78005, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5127, 78132, F5, 23) (dual of [78132, 78005, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(5127, 78125, F5, 23) (dual of [78125, 77998, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(50, 7, F5, 0) (dual of [7, 7, 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(22) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(5127, 78132, F5, 23) (dual of [78132, 78005, 24]-code), using
(104, 104+23, large)-Net in Base 5 — Upper bound on s
There is no (104, 127, large)-net in base 5, because
- 21 times m-reduction [i] would yield (104, 106, large)-net in base 5, but