Best Known (71, 88, s)-Nets in Base 4
(71, 88, 2050)-Net over F4 — Constructive and digital
Digital (71, 88, 2050)-net over F4, using
- net defined by OOA [i] based on linear OOA(488, 2050, F4, 17, 17) (dual of [(2050, 17), 34762, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- 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(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
(71, 88, 8200)-Net over F4 — Digital
Digital (71, 88, 8200)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(488, 8200, F4, 2, 17) (dual of [(8200, 2), 16312, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(488, 16400, F4, 17) (dual of [16400, 16312, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- 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(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(16) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
- OOA 2-folding [i] based on linear OA(488, 16400, F4, 17) (dual of [16400, 16312, 18]-code), using
(71, 88, 4425584)-Net in Base 4 — Upper bound on s
There is no (71, 88, 4425585)-net in base 4, because
- 1 times m-reduction [i] would yield (71, 87, 4425585)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 23945 275991 760676 704649 766094 708342 403250 058342 561562 > 487 [i]