Best Known (128−17, 128, s)-Nets in Base 4
(128−17, 128, 131076)-Net over F4 — Constructive and digital
Digital (111, 128, 131076)-net over F4, using
- 42 times duplication [i] based on digital (109, 126, 131076)-net over F4, using
- net defined by OOA [i] based on linear OOA(4126, 131076, F4, 17, 17) (dual of [(131076, 17), 2228166, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4126, 1048609, F4, 17) (dual of [1048609, 1048483, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4126, 1048609, F4, 17) (dual of [1048609, 1048483, 18]-code), using
- net defined by OOA [i] based on linear OOA(4126, 131076, F4, 17, 17) (dual of [(131076, 17), 2228166, 18]-NRT-code), using
(128−17, 128, 524306)-Net over F4 — Digital
Digital (111, 128, 524306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4128, 524306, F4, 2, 17) (dual of [(524306, 2), 1048484, 18]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4127, 524306, F4, 2, 17) (dual of [(524306, 2), 1048485, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4127, 1048612, F4, 17) (dual of [1048612, 1048485, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(4121, 1048576, F4, 17) (dual of [1048576, 1048455, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(491, 1048576, F4, 13) (dual of [1048576, 1048485, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(16) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4126, 1048611, F4, 17) (dual of [1048611, 1048485, 18]-code), using
- OOA 2-folding [i] based on linear OA(4127, 1048612, F4, 17) (dual of [1048612, 1048485, 18]-code), using
- 41 times duplication [i] based on linear OOA(4127, 524306, F4, 2, 17) (dual of [(524306, 2), 1048485, 18]-NRT-code), using
(128−17, 128, large)-Net in Base 4 — Upper bound on s
There is no (111, 128, large)-net in base 4, because
- 15 times m-reduction [i] would yield (111, 113, large)-net in base 4, but