Best Known (128−23, 128, s)-Nets in Base 5
(128−23, 128, 7103)-Net over F5 — Constructive and digital
Digital (105, 128, 7103)-net over F5, using
- net defined by OOA [i] based on linear OOA(5128, 7103, F5, 23, 23) (dual of [(7103, 23), 163241, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5128, 78134, F5, 23) (dual of [78134, 78006, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 78141, F5, 23) (dual of [78141, 78013, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5127, 78126, F5, 23) (dual of [78126, 77999, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(5113, 78126, F5, 21) (dual of [78126, 78013, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5128, 78141, F5, 23) (dual of [78141, 78013, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(5128, 78134, F5, 23) (dual of [78134, 78006, 24]-code), using
(128−23, 128, 39070)-Net over F5 — Digital
Digital (105, 128, 39070)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5128, 39070, F5, 2, 23) (dual of [(39070, 2), 78012, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5128, 78140, F5, 23) (dual of [78140, 78012, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 78141, F5, 23) (dual of [78141, 78013, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(5127, 78126, F5, 23) (dual of [78126, 77999, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(5113, 78126, F5, 21) (dual of [78126, 78013, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 78126 | 514−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(51, 15, F5, 1) (dual of [15, 14, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5128, 78141, F5, 23) (dual of [78141, 78013, 24]-code), using
- OOA 2-folding [i] based on linear OA(5128, 78140, F5, 23) (dual of [78140, 78012, 24]-code), using
(128−23, 128, large)-Net in Base 5 — Upper bound on s
There is no (105, 128, large)-net in base 5, because
- 21 times m-reduction [i] would yield (105, 107, large)-net in base 5, but