Best Known (193, 193+29, s)-Nets in Base 4
(193, 193+29, 74901)-Net over F4 — Constructive and digital
Digital (193, 222, 74901)-net over F4, using
- 42 times duplication [i] based on digital (191, 220, 74901)-net over F4, using
- net defined by OOA [i] based on linear OOA(4220, 74901, F4, 29, 29) (dual of [(74901, 29), 2171909, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4220, 1048615, F4, 29) (dual of [1048615, 1048395, 30]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4216, 1048611, F4, 29) (dual of [1048611, 1048395, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4220, 1048615, F4, 29) (dual of [1048615, 1048395, 30]-code), using
- net defined by OOA [i] based on linear OOA(4220, 74901, F4, 29, 29) (dual of [(74901, 29), 2171909, 30]-NRT-code), using
(193, 193+29, 437024)-Net over F4 — Digital
Digital (193, 222, 437024)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4222, 437024, F4, 2, 29) (dual of [(437024, 2), 873826, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4222, 524313, F4, 2, 29) (dual of [(524313, 2), 1048404, 30]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4221, 524313, F4, 2, 29) (dual of [(524313, 2), 1048405, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4221, 1048626, F4, 29) (dual of [1048626, 1048405, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 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(22) [i] based on
- OOA 2-folding [i] based on linear OA(4221, 1048626, F4, 29) (dual of [1048626, 1048405, 30]-code), using
- 41 times duplication [i] based on linear OOA(4221, 524313, F4, 2, 29) (dual of [(524313, 2), 1048405, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4222, 524313, F4, 2, 29) (dual of [(524313, 2), 1048404, 30]-NRT-code), using
(193, 193+29, large)-Net in Base 4 — Upper bound on s
There is no (193, 222, large)-net in base 4, because
- 27 times m-reduction [i] would yield (193, 195, large)-net in base 4, but