Best Known (29, 52, s)-Nets in Base 128
(29, 52, 1491)-Net over F128 — Constructive and digital
Digital (29, 52, 1491)-net over F128, using
- 1282 times duplication [i] based on digital (27, 50, 1491)-net over F128, using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12833, 16385, F128, 17) (dual of [16385, 16352, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,11]) ⊂ C([0,8]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(12850, 16402, F128, 23) (dual of [16402, 16352, 24]-code), using
- net defined by OOA [i] based on linear OOA(12850, 1491, F128, 23, 23) (dual of [(1491, 23), 34243, 24]-NRT-code), using
(29, 52, 5957)-Net in Base 128 — Constructive
(29, 52, 5957)-net in base 128, using
- net defined by OOA [i] based on OOA(12852, 5957, S128, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12852, 65528, S128, 23), using
- discarding factors based on OA(12852, 65538, S128, 23), using
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(25645, 65536, F256, 23) (dual of [65536, 65491, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(25643, 65536, F256, 22) (dual of [65536, 65493, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding parts of the base [i] based on linear OA(25645, 65538, F256, 23) (dual of [65538, 65493, 24]-code), using
- discarding factors based on OA(12852, 65538, S128, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12852, 65528, S128, 23), using
(29, 52, 8948)-Net over F128 — Digital
Digital (29, 52, 8948)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12852, 8948, F128, 23) (dual of [8948, 8896, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12852, 16408, F128, 23) (dual of [16408, 16356, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- linear OA(12845, 16385, F128, 23) (dual of [16385, 16340, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(1287, 23, F128, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,128)), using
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- Reed–Solomon code RS(121,128) [i]
- discarding factors / shortening the dual code based on linear OA(1287, 128, F128, 7) (dual of [128, 121, 8]-code or 128-arc in PG(6,128)), using
- construction X applied to C([0,11]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12852, 16408, F128, 23) (dual of [16408, 16356, 24]-code), using
(29, 52, large)-Net in Base 128 — Upper bound on s
There is no (29, 52, large)-net in base 128, because
- 21 times m-reduction [i] would yield (29, 31, large)-net in base 128, but