MinT - Best Known (105−17, 105, s)-Nets in Base 32

Best Known (105−17, 105, s)-Nets in Base 32

(105−17, 105, 1056769)-Net over F₃₂ — Constructive and digital

Digital (88, 105, 1056769)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (16, 24, 8194)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32²⁴, 8194, F₃₂, 8, 8) (dual of [(8194, 8), 65528, 9]-NRT-code), using
    - OA 4-folding and stacking [i] based on linear OA(32²⁴, 32776, F₃₂, 8) (dual of [32776, 32752, 9]-code), using
      - discarding factors / shortening the dual code based on linear OA(32²⁴, 32779, F₃₂, 8) (dual of [32779, 32755, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(4) [i] based on
        linear OA(32²², 32768, F₃₂, 8) (dual of [32768, 32746, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(32¹³, 32768, F₃₂, 5) (dual of [32768, 32755, 6]-code), using an extension C_e(4) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
        linear OA(32², 11, F₃₂, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,32)), using
        discarding factors / shortening the dual code based on linear OA(32², 32, F₃₂, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
        Reedâ€“Solomon code RS(30,32) [i]
2. digital (64, 81, 1048575)-net over F₃₂, using
  - net defined by OOA [i] based on linear OOA(32⁸¹, 1048575, F₃₂, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
    - OOA 8-folding and stacking with additional row [i] based on linear OA(32⁸¹, 8388601, F₃₂, 17) (dual of [8388601, 8388520, 18]-code), using
      - discarding factors / shortening the dual code based on linear OA(32⁸¹, large, F₃₂, 17) (dual of [large, large−81, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 33554433Â | 32¹⁰âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(105−17, 105, 1114111)-Net in Base 32 — Constructive

(88, 105, 1114111)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (19, 27, 65536)-net in base 32, using
  - net defined by OOA [i] based on OOA(32²⁷, 65536, S₃₂, 8, 8), using
    - OA 4-folding and stacking [i] based on OA(32²⁷, 262144, S₃₂, 8), using
      - discarding factors based on OA(32²⁷, 262147, S₃₂, 8), using
        discarding parts of the base [i] based on linear OA(64²², 262147, F₆₄, 8) (dual of [262147, 262125, 9]-code), using
        constructionÂ X applied to C_e(7) âŠ‚ C_e(6) [i] based on
        linear OA(64²², 262144, F₆₄, 8) (dual of [262144, 262122, 9]-code), using an extension C_e(7) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,7], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]
        linear OA(64¹⁹, 262144, F₆₄, 7) (dual of [262144, 262125, 8]-code), using an extension C_e(6) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 64³âˆ’1, defining interval IÂ =Â [1,6], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 7 [i]
        linear OA(64⁰, 3, F₆₄, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]
2. (61, 78, 1048575)-net in base 32, using
  - base change [i] based on digital (48, 65, 1048575)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁶⁵, 1048575, F₆₄, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
      - OOA 8-folding and stacking with additional row [i] based on linear OA(64⁶⁵, 8388601, F₆₄, 17) (dual of [8388601, 8388536, 18]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁶⁵, large, F₆₄, 17) (dual of [large, large−65, 18]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]

(105−17, 105, large)-Net over F₃₂ — Digital

Digital (88, 105, large)-net over F₃₂, using

t-expansion [i] based on digital (87, 105, large)-net over F₃₂, using
- 5 times m-reduction [i] based on digital (87, 110, large)-net over F₃₂, using
  - Gilbertâ€“VarÅ¡amov bound [i]

(105−17, 105, large)-Net in Base 32 — Upper bound on s

There is no (88, 105, large)-net in base 32, because

15 times m-reduction [i] would yield (88, 90, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]

Best Known (105−17, 105, s)-Nets in Base 32

(105−17, 105, 1056769)-Net over F32 — Constructive and digital

(105−17, 105, 1114111)-Net in Base 32 — Constructive

(105−17, 105, large)-Net over F32 — Digital

(105−17, 105, large)-Net in Base 32 — Upper bound on s

(105−17, 105, 1056769)-Net over F₃₂ — Constructive and digital

(105−17, 105, large)-Net over F₃₂ — Digital