Best Known (107, 107+20, s)-Nets in Base 4
(107, 107+20, 6556)-Net over F4 — Constructive and digital
Digital (107, 127, 6556)-net over F4, using
- 41 times duplication [i] based on digital (106, 126, 6556)-net over F4, using
- t-expansion [i] based on digital (105, 126, 6556)-net over F4, using
- net defined by OOA [i] based on linear OOA(4126, 6556, F4, 21, 21) (dual of [(6556, 21), 137550, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4126, 65561, F4, 21) (dual of [65561, 65435, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-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(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4126, 65565, F4, 21) (dual of [65565, 65439, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4126, 65561, F4, 21) (dual of [65561, 65435, 22]-code), using
- net defined by OOA [i] based on linear OOA(4126, 6556, F4, 21, 21) (dual of [(6556, 21), 137550, 22]-NRT-code), using
- t-expansion [i] based on digital (105, 126, 6556)-net over F4, using
(107, 107+20, 41236)-Net over F4 — Digital
Digital (107, 127, 41236)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4127, 41236, F4, 20) (dual of [41236, 41109, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4127, 65567, F4, 20) (dual of [65567, 65440, 21]-code), using
- 1 times truncation [i] based on linear OA(4128, 65568, F4, 21) (dual of [65568, 65440, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4121, 65536, F4, 21) (dual of [65536, 65415, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(497, 65536, F4, 17) (dual of [65536, 65439, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(4128, 65568, F4, 21) (dual of [65568, 65440, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4127, 65567, F4, 20) (dual of [65567, 65440, 21]-code), using
(107, 107+20, large)-Net in Base 4 — Upper bound on s
There is no (107, 127, large)-net in base 4, because
- 18 times m-reduction [i] would yield (107, 109, large)-net in base 4, but