Best Known (193−28, 193, s)-Nets in Base 4
(193−28, 193, 18726)-Net over F4 — Constructive and digital
Digital (165, 193, 18726)-net over F4, using
- t-expansion [i] based on digital (164, 193, 18726)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 18726, F4, 29, 29) (dual of [(18726, 29), 542861, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(4193, 262165, F4, 29) (dual of [262165, 261972, 30]-code), using
- net defined by OOA [i] based on linear OOA(4193, 18726, F4, 29, 29) (dual of [(18726, 29), 542861, 30]-NRT-code), using
(193−28, 193, 131083)-Net over F4 — Digital
Digital (165, 193, 131083)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4193, 131083, F4, 2, 28) (dual of [(131083, 2), 261973, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4193, 262166, F4, 28) (dual of [262166, 261973, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 262167, F4, 28) (dual of [262167, 261974, 29]-code), using
- construction XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4172, 262144, F4, 26) (dual of [262144, 261972, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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 XX applied to Ce(28) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4193, 262167, F4, 28) (dual of [262167, 261974, 29]-code), using
- OOA 2-folding [i] based on linear OA(4193, 262166, F4, 28) (dual of [262166, 261973, 29]-code), using
(193−28, 193, large)-Net in Base 4 — Upper bound on s
There is no (165, 193, large)-net in base 4, because
- 26 times m-reduction [i] would yield (165, 167, large)-net in base 4, but