Best Known (117−23, 117, s)-Nets in Base 5
(117−23, 117, 1423)-Net over F5 — Constructive and digital
Digital (94, 117, 1423)-net over F5, using
- 52 times duplication [i] based on digital (92, 115, 1423)-net over F5, using
- net defined by OOA [i] based on linear OOA(5115, 1423, F5, 23, 23) (dual of [(1423, 23), 32614, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5115, 15654, F5, 23) (dual of [15654, 15539, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5115, 15655, F5, 23) (dual of [15655, 15540, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- linear OA(5109, 15625, F5, 23) (dual of [15625, 15516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(585, 15625, F5, 18) (dual of [15625, 15540, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(22) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5115, 15655, F5, 23) (dual of [15655, 15540, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5115, 15654, F5, 23) (dual of [15654, 15539, 24]-code), using
- net defined by OOA [i] based on linear OOA(5115, 1423, F5, 23, 23) (dual of [(1423, 23), 32614, 24]-NRT-code), using
(117−23, 117, 15658)-Net over F5 — Digital
Digital (94, 117, 15658)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5117, 15658, F5, 23) (dual of [15658, 15541, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(5109, 15626, F5, 23) (dual of [15626, 15517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(585, 15626, F5, 17) (dual of [15626, 15541, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(58, 32, F5, 5) (dual of [32, 24, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(58, 33, F5, 5) (dual of [33, 25, 6]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
(117−23, 117, large)-Net in Base 5 — Upper bound on s
There is no (94, 117, large)-net in base 5, because
- 21 times m-reduction [i] would yield (94, 96, large)-net in base 5, but