Best Known (127−19, 127, s)-Nets in Base 4
(127−19, 127, 29128)-Net over F4 — Constructive and digital
Digital (108, 127, 29128)-net over F4, using
- net defined by OOA [i] based on linear OOA(4127, 29128, F4, 19, 19) (dual of [(29128, 19), 553305, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4127, 262153, F4, 19) (dual of [262153, 262026, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4127, 262144, F4, 19) (dual of [262144, 262017, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(4127, 262153, F4, 19) (dual of [262153, 262026, 20]-code), using
(127−19, 127, 114542)-Net over F4 — Digital
Digital (108, 127, 114542)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4127, 114542, F4, 2, 19) (dual of [(114542, 2), 228957, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4127, 131076, F4, 2, 19) (dual of [(131076, 2), 262025, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4127, 262152, F4, 19) (dual of [262152, 262025, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4127, 262153, F4, 19) (dual of [262153, 262026, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4127, 262144, F4, 19) (dual of [262144, 262017, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4127, 262153, F4, 19) (dual of [262153, 262026, 20]-code), using
- OOA 2-folding [i] based on linear OA(4127, 262152, F4, 19) (dual of [262152, 262025, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(4127, 131076, F4, 2, 19) (dual of [(131076, 2), 262025, 20]-NRT-code), using
(127−19, 127, large)-Net in Base 4 — Upper bound on s
There is no (108, 127, large)-net in base 4, because
- 17 times m-reduction [i] would yield (108, 110, large)-net in base 4, but