Best Known (207−27, 207, s)-Nets in Base 4
(207−27, 207, 80662)-Net over F4 — Constructive and digital
Digital (180, 207, 80662)-net over F4, using
- 41 times duplication [i] based on digital (179, 206, 80662)-net over F4, using
- net defined by OOA [i] based on linear OOA(4206, 80662, F4, 27, 27) (dual of [(80662, 27), 2177668, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4206, 1048607, F4, 27) (dual of [1048607, 1048401, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4206, 1048611, F4, 27) (dual of [1048611, 1048405, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(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(26) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4206, 1048611, F4, 27) (dual of [1048611, 1048405, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4206, 1048607, F4, 27) (dual of [1048607, 1048401, 28]-code), using
- net defined by OOA [i] based on linear OOA(4206, 80662, F4, 27, 27) (dual of [(80662, 27), 2177668, 28]-NRT-code), using
(207−27, 207, 453753)-Net over F4 — Digital
Digital (180, 207, 453753)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4207, 453753, F4, 2, 27) (dual of [(453753, 2), 907299, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4207, 524306, F4, 2, 27) (dual of [(524306, 2), 1048405, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4207, 1048612, F4, 27) (dual of [1048612, 1048405, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4206, 1048611, F4, 27) (dual of [1048611, 1048405, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(22) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(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(26) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4206, 1048611, F4, 27) (dual of [1048611, 1048405, 28]-code), using
- OOA 2-folding [i] based on linear OA(4207, 1048612, F4, 27) (dual of [1048612, 1048405, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(4207, 524306, F4, 2, 27) (dual of [(524306, 2), 1048405, 28]-NRT-code), using
(207−27, 207, large)-Net in Base 4 — Upper bound on s
There is no (180, 207, large)-net in base 4, because
- 25 times m-reduction [i] would yield (180, 182, large)-net in base 4, but