Best Known (32, 32+23, s)-Nets in Base 128
(32, 32+23, 1492)-Net over F128 — Constructive and digital
Digital (32, 55, 1492)-net over F128, using
- 1281 times duplication [i] based on digital (31, 54, 1492)-net over F128, using
- net defined by OOA [i] based on linear OOA(12854, 1492, F128, 23, 23) (dual of [(1492, 23), 34262, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12854, 16413, F128, 23) (dual of [16413, 16359, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(12854, 16414, F128, 23) (dual of [16414, 16360, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [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(12825, 16385, F128, 13) (dual of [16385, 16360, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(1289, 29, F128, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,128)), using
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- Reed–Solomon code RS(119,128) [i]
- discarding factors / shortening the dual code based on linear OA(1289, 128, F128, 9) (dual of [128, 119, 10]-code or 128-arc in PG(8,128)), using
- construction X applied to C([0,11]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(12854, 16414, F128, 23) (dual of [16414, 16360, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(12854, 16413, F128, 23) (dual of [16413, 16359, 24]-code), using
- net defined by OOA [i] based on linear OOA(12854, 1492, F128, 23, 23) (dual of [(1492, 23), 34262, 24]-NRT-code), using
(32, 32+23, 5958)-Net in Base 128 — Constructive
(32, 55, 5958)-net in base 128, using
- 1282 times duplication [i] based on (30, 53, 5958)-net in base 128, using
- net defined by OOA [i] based on OOA(12853, 5958, S128, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12853, 65539, S128, 23), using
- discarding factors based on OA(12853, 65542, S128, 23), using
- discarding parts of the base [i] based on linear OA(25646, 65542, F256, 23) (dual of [65542, 65496, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 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 parts of the base [i] based on linear OA(25646, 65542, F256, 23) (dual of [65542, 65496, 24]-code), using
- discarding factors based on OA(12853, 65542, S128, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(12853, 65539, S128, 23), using
- net defined by OOA [i] based on OOA(12853, 5958, S128, 23, 23), using
(32, 32+23, 16416)-Net over F128 — Digital
Digital (32, 55, 16416)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12855, 16416, F128, 23) (dual of [16416, 16361, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
- linear OA(12845, 16384, F128, 23) (dual of [16384, 16339, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12810, 32, F128, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,128)), using
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- Reed–Solomon code RS(118,128) [i]
- discarding factors / shortening the dual code based on linear OA(12810, 128, F128, 10) (dual of [128, 118, 11]-code or 128-arc in PG(9,128)), using
- construction X applied to Ce(22) ⊂ Ce(11) [i] based on
(32, 32+23, large)-Net in Base 128 — Upper bound on s
There is no (32, 55, large)-net in base 128, because
- 21 times m-reduction [i] would yield (32, 34, large)-net in base 128, but