Best Known (100−21, 100, s)-Nets in Base 5
(100−21, 100, 1563)-Net over F5 — Constructive and digital
Digital (79, 100, 1563)-net over F5, using
- 52 times duplication [i] based on digital (77, 98, 1563)-net over F5, using
- net defined by OOA [i] based on linear OOA(598, 1563, F5, 21, 21) (dual of [(1563, 21), 32725, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(598, 15631, F5, 21) (dual of [15631, 15533, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(598, 15632, F5, 21) (dual of [15632, 15534, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(597, 15625, F5, 21) (dual of [15625, 15528, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(591, 15625, F5, 19) (dual of [15625, 15534, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(51, 7, F5, 1) (dual of [7, 6, 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 Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(598, 15632, F5, 21) (dual of [15632, 15534, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(598, 15631, F5, 21) (dual of [15631, 15533, 22]-code), using
- net defined by OOA [i] based on linear OOA(598, 1563, F5, 21, 21) (dual of [(1563, 21), 32725, 22]-NRT-code), using
(100−21, 100, 8680)-Net over F5 — Digital
Digital (79, 100, 8680)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5100, 8680, F5, 21) (dual of [8680, 8580, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5100, 15632, F5, 21) (dual of [15632, 15532, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(597, 15626, F5, 21) (dual of [15626, 15529, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (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(53, 6, F5, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,5) or 6-cap in PG(2,5)), using
- extended Reed–Solomon code RSe(3,5) [i]
- oval in PG(2, 5) [i]
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(5100, 15632, F5, 21) (dual of [15632, 15532, 22]-code), using
(100−21, 100, large)-Net in Base 5 — Upper bound on s
There is no (79, 100, large)-net in base 5, because
- 19 times m-reduction [i] would yield (79, 81, large)-net in base 5, but