Best Known (113−15, 113, s)-Nets in Base 5
(113−15, 113, 279022)-Net over F5 — Constructive and digital
Digital (98, 113, 279022)-net over F5, using
- 51 times duplication [i] based on digital (97, 112, 279022)-net over F5, using
- net defined by OOA [i] based on linear OOA(5112, 279022, F5, 15, 15) (dual of [(279022, 15), 4185218, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(5112, 1953155, F5, 15) (dual of [1953155, 1953043, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(5109, 1953125, F5, 16) (dual of [1953125, 1953016, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(582, 1953125, F5, 12) (dual of [1953125, 1953043, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(53, 30, F5, 2) (dual of [30, 27, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(5112, 1953155, F5, 15) (dual of [1953155, 1953043, 16]-code), using
- net defined by OOA [i] based on linear OOA(5112, 279022, F5, 15, 15) (dual of [(279022, 15), 4185218, 16]-NRT-code), using
(113−15, 113, 1490181)-Net over F5 — Digital
Digital (98, 113, 1490181)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5113, 1490181, F5, 15) (dual of [1490181, 1490068, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 1953152, F5, 15) (dual of [1953152, 1953039, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(5109, 1953126, F5, 15) (dual of [1953126, 1953017, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(573, 1953126, F5, 11) (dual of [1953126, 1953053, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5113, 1953152, F5, 15) (dual of [1953152, 1953039, 16]-code), using
(113−15, 113, large)-Net in Base 5 — Upper bound on s
There is no (98, 113, large)-net in base 5, because
- 13 times m-reduction [i] would yield (98, 100, large)-net in base 5, but