Best Known (44, 54, s)-Nets in Base 4
(44, 54, 3280)-Net over F4 — Constructive and digital
Digital (44, 54, 3280)-net over F4, using
- 41 times duplication [i] based on digital (43, 53, 3280)-net over F4, using
- net defined by OOA [i] based on linear OOA(453, 3280, F4, 10, 10) (dual of [(3280, 10), 32747, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(453, 16400, F4, 10) (dual of [16400, 16347, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(453, 16401, F4, 10) (dual of [16401, 16348, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(453, 16401, F4, 10) (dual of [16401, 16348, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(453, 16400, F4, 10) (dual of [16400, 16347, 11]-code), using
- net defined by OOA [i] based on linear OOA(453, 3280, F4, 10, 10) (dual of [(3280, 10), 32747, 11]-NRT-code), using
(44, 54, 12218)-Net over F4 — Digital
Digital (44, 54, 12218)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(454, 12218, F4, 10) (dual of [12218, 12164, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(454, 16403, F4, 10) (dual of [16403, 16349, 11]-code), using
- construction XX applied to Ce(9) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(9) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(454, 16403, F4, 10) (dual of [16403, 16349, 11]-code), using
(44, 54, 2760338)-Net in Base 4 — Upper bound on s
There is no (44, 54, 2760339)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 324 518918 080519 894117 419053 978680 > 454 [i]