Best Known (54, 54+13, s)-Nets in Base 4
(54, 54+13, 2733)-Net over F4 — Constructive and digital
Digital (54, 67, 2733)-net over F4, using
- net defined by OOA [i] based on linear OOA(467, 2733, F4, 13, 13) (dual of [(2733, 13), 35462, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(467, 16399, F4, 13) (dual of [16399, 16332, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(467, 16401, F4, 13) (dual of [16401, 16334, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(467, 16401, F4, 13) (dual of [16401, 16334, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(467, 16399, F4, 13) (dual of [16399, 16332, 14]-code), using
(54, 54+13, 8200)-Net over F4 — Digital
Digital (54, 67, 8200)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(467, 8200, F4, 2, 13) (dual of [(8200, 2), 16333, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(467, 16400, F4, 13) (dual of [16400, 16333, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(467, 16401, F4, 13) (dual of [16401, 16334, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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(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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(467, 16401, F4, 13) (dual of [16401, 16334, 14]-code), using
- OOA 2-folding [i] based on linear OA(467, 16400, F4, 13) (dual of [16400, 16333, 14]-code), using
(54, 54+13, 4185624)-Net in Base 4 — Upper bound on s
There is no (54, 67, 4185625)-net in base 4, because
- 1 times m-reduction [i] would yield (54, 66, 4185625)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 5444 521674 318891 417677 399510 328817 783876 > 466 [i]