Best Known (55, 55+13, s)-Nets in Base 4
(55, 55+13, 2733)-Net over F4 — Constructive and digital
Digital (55, 68, 2733)-net over F4, using
- 41 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(467, 2733, F4, 13, 13) (dual of [(2733, 13), 35462, 14]-NRT-code), using
(55, 55+13, 8201)-Net over F4 — Digital
Digital (55, 68, 8201)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(468, 8201, F4, 2, 13) (dual of [(8201, 2), 16334, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(468, 16402, F4, 13) (dual of [16402, 16334, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(468, 16403, F4, 13) (dual of [16403, 16335, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(9) ⊂ Ce(8) [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(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(12) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(468, 16403, F4, 13) (dual of [16403, 16335, 14]-code), using
- OOA 2-folding [i] based on linear OA(468, 16402, F4, 13) (dual of [16402, 16334, 14]-code), using
(55, 55+13, 5273557)-Net in Base 4 — Upper bound on s
There is no (55, 68, 5273558)-net in base 4, because
- 1 times m-reduction [i] would yield (55, 67, 5273558)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 21778 081392 049894 443940 846297 910596 848880 > 467 [i]