Best Known (113−20, 113, s)-Nets in Base 4
(113−20, 113, 1641)-Net over F4 — Constructive and digital
Digital (93, 113, 1641)-net over F4, using
- 41 times duplication [i] based on digital (92, 112, 1641)-net over F4, using
- t-expansion [i] based on digital (91, 112, 1641)-net over F4, using
- net defined by OOA [i] based on linear OOA(4112, 1641, F4, 21, 21) (dual of [(1641, 21), 34349, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4112, 16411, F4, 21) (dual of [16411, 16299, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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
- 1 times code embedding in larger space [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4112, 16411, F4, 21) (dual of [16411, 16299, 22]-code), using
- net defined by OOA [i] based on linear OOA(4112, 1641, F4, 21, 21) (dual of [(1641, 21), 34349, 22]-NRT-code), using
- t-expansion [i] based on digital (91, 112, 1641)-net over F4, using
(113−20, 113, 14019)-Net over F4 — Digital
Digital (93, 113, 14019)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4113, 14019, F4, 20) (dual of [14019, 13906, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 16419, F4, 20) (dual of [16419, 16306, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4113, 16419, F4, 20) (dual of [16419, 16306, 21]-code), using
(113−20, 113, large)-Net in Base 4 — Upper bound on s
There is no (93, 113, large)-net in base 4, because
- 18 times m-reduction [i] would yield (93, 95, large)-net in base 4, but