Best Known (172, 172+28, s)-Nets in Base 4
(172, 172+28, 18728)-Net over F4 — Constructive and digital
Digital (172, 200, 18728)-net over F4, using
- net defined by OOA [i] based on linear OOA(4200, 18728, F4, 28, 28) (dual of [(18728, 28), 524184, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(4200, 262192, F4, 28) (dual of [262192, 261992, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4200, 262199, F4, 28) (dual of [262199, 261999, 29]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [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(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(410, 55, F4, 5) (dual of [55, 45, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4200, 262199, F4, 28) (dual of [262199, 261999, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(4200, 262192, F4, 28) (dual of [262192, 261992, 29]-code), using
(172, 172+28, 142621)-Net over F4 — Digital
Digital (172, 200, 142621)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4200, 142621, F4, 28) (dual of [142621, 142421, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(4200, 262173, F4, 28) (dual of [262173, 261973, 29]-code), using
- construction X4 applied to Ce(29) ⊂ Ce(25) [i] based on
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(428, 29, F4, 28) (dual of [29, 1, 29]-code or 29-arc in PG(27,4)), using
- dual of repetition code with length 29 [i]
- linear OA(41, 29, F4, 1) (dual of [29, 28, 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
- construction X4 applied to Ce(29) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4200, 262173, F4, 28) (dual of [262173, 261973, 29]-code), using
(172, 172+28, large)-Net in Base 4 — Upper bound on s
There is no (172, 200, large)-net in base 4, because
- 26 times m-reduction [i] would yield (172, 174, large)-net in base 4, but