MinT - Best Known (78, 78+25, s)-Nets in Base 16

Best Known (78, 78+25, s)-Nets in Base 16

(78, 78+25, 10924)-Net over F₁₆ — Constructive and digital

Digital (78, 103, 10924)-net over F₁₆, using

net defined by OOA [i] based on linear OOA(16¹⁰³, 10924, F₁₆, 25, 25) (dual of [(10924, 25), 272997, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(16¹⁰³, 131089, F₁₆, 25) (dual of [131089, 130986, 26]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(16¹⁰², 131088, F₁₆, 25) (dual of [131088, 130986, 26]-code), using
    - trace code [i] based on linear OA(256⁵¹, 65544, F₂₅₆, 25) (dual of [65544, 65493, 26]-code), using
      - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
        linear OA(256⁴⁹, 65536, F₂₅₆, 25) (dual of [65536, 65487, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(256⁴³, 65536, F₂₅₆, 22) (dual of [65536, 65493, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(256², 8, F₂₅₆, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
        discarding factors / shortening the dual code based on linear OA(256², 256, F₂₅₆, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
        Reedâ€“Solomon code RS(254,256) [i]

(78, 78+25, 131090)-Net over F₁₆ — Digital

Digital (78, 103, 131090)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰³, 131090, F₁₆, 25) (dual of [131090, 130987, 26]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(16¹⁰², 131088, F₁₆, 25) (dual of [131088, 130986, 26]-code), using
    - trace code [i] based on linear OA(256⁵¹, 65544, F₂₅₆, 25) (dual of [65544, 65493, 26]-code), using
      - constructionÂ X applied to C_e(24) âŠ‚ C_e(21) [i] based on
        linear OA(256⁴⁹, 65536, F₂₅₆, 25) (dual of [65536, 65487, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
        linear OA(256⁴³, 65536, F₂₅₆, 22) (dual of [65536, 65493, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 256²âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
        linear OA(256², 8, F₂₅₆, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
        discarding factors / shortening the dual code based on linear OA(256², 256, F₂₅₆, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
        Reedâ€“Solomon code RS(254,256) [i]
  2. linear OA(16¹⁰², 131089, F₁₆, 24) (dual of [131089, 130987, 25]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 16¹⁰²Â > V_b^sâˆ’1(kâˆ’1)Â = 219 138627 191097 057956 555815 242398 947382 803458 196104 678552 366205 430529 459546 682521 258284 738191 702926 089881 315075 876094 058496 [i]
  3. linear OA(16⁰, 1, F₁₆, 0) (dual of [1, 1, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(78, 78+25, large)-Net in Base 16 — Upper bound on s

There is no (78, 103, large)-net in base 16, because

23 times m-reduction [i] would yield (78, 80, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (78, 78+25, s)-Nets in Base 16

(78, 78+25, 10924)-Net over F16 — Constructive and digital

(78, 78+25, 131090)-Net over F16 — Digital

(78, 78+25, large)-Net in Base 16 — Upper bound on s

(78, 78+25, 10924)-Net over F₁₆ — Constructive and digital

(78, 78+25, 131090)-Net over F₁₆ — Digital