Best Known (110−17, 110, s)-Nets in Base 4
(110−17, 110, 32769)-Net over F4 — Constructive and digital
Digital (93, 110, 32769)-net over F4, using
- net defined by OOA [i] based on linear OOA(4110, 32769, F4, 17, 17) (dual of [(32769, 17), 556963, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4110, 262153, F4, 17) (dual of [262153, 262043, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 262154, F4, 17) (dual of [262154, 262044, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(4110, 262154, F4, 17) (dual of [262154, 262044, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(4110, 262153, F4, 17) (dual of [262153, 262043, 18]-code), using
(110−17, 110, 88868)-Net over F4 — Digital
Digital (93, 110, 88868)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4110, 88868, F4, 2, 17) (dual of [(88868, 2), 177626, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4110, 131077, F4, 2, 17) (dual of [(131077, 2), 262044, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4110, 262154, F4, 17) (dual of [262154, 262044, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(4109, 262144, F4, 17) (dual of [262144, 262035, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(4100, 262144, F4, 15) (dual of [262144, 262044, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(41, 10, F4, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- OOA 2-folding [i] based on linear OA(4110, 262154, F4, 17) (dual of [262154, 262044, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(4110, 131077, F4, 2, 17) (dual of [(131077, 2), 262044, 18]-NRT-code), using
(110−17, 110, large)-Net in Base 4 — Upper bound on s
There is no (93, 110, large)-net in base 4, because
- 15 times m-reduction [i] would yield (93, 95, large)-net in base 4, but