Best Known (89, 89+21, s)-Nets in Base 4
(89, 89+21, 1640)-Net over F4 — Constructive and digital
Digital (89, 110, 1640)-net over F4, using
- 41 times duplication [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
(89, 89+21, 8201)-Net over F4 — Digital
Digital (89, 110, 8201)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4110, 8201, F4, 2, 21) (dual of [(8201, 2), 16292, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4110, 16402, F4, 21) (dual of [16402, 16292, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 16403, F4, 21) (dual of [16403, 16293, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ 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(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(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(43, 18, F4, 2) (dual of [18, 15, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(4110, 16403, F4, 21) (dual of [16403, 16293, 22]-code), using
- OOA 2-folding [i] based on linear OA(4110, 16402, F4, 21) (dual of [16402, 16292, 22]-code), using
(89, 89+21, 5511988)-Net in Base 4 — Upper bound on s
There is no (89, 110, 5511989)-net in base 4, because
- 1 times m-reduction [i] would yield (89, 109, 5511989)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 421249 218590 287845 864618 287883 951718 756513 430066 154184 401864 713308 > 4109 [i]