Best Known (90, 110, s)-Nets in Base 4
(90, 110, 1640)-Net over F4 — Constructive and digital
Digital (90, 110, 1640)-net over F4, using
- 41 times duplication [i] based on digital (89, 109, 1640)-net over F4, using
- t-expansion [i] based on digital (88, 109, 1640)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 1640, F4, 21, 21) (dual of [(1640, 21), 34331, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4109, 16401, F4, 21) (dual of [16401, 16292, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [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(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(4109, 16401, F4, 21) (dual of [16401, 16292, 22]-code), using
- net defined by OOA [i] based on linear OOA(4109, 1640, F4, 21, 21) (dual of [(1640, 21), 34331, 22]-NRT-code), using
- t-expansion [i] based on digital (88, 109, 1640)-net over F4, using
(90, 110, 11124)-Net over F4 — Digital
Digital (90, 110, 11124)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4110, 11124, F4, 20) (dual of [11124, 11014, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 16409, F4, 20) (dual of [16409, 16299, 21]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(4111, 16410, F4, 21) (dual of [16410, 16299, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 16409, F4, 20) (dual of [16409, 16299, 21]-code), using
(90, 110, 6331613)-Net in Base 4 — Upper bound on s
There is no (90, 110, 6331614)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 684997 241973 480177 027741 543866 136752 257842 491914 459424 129523 941058 > 4110 [i]