November  2016, 10(4): 847-850. doi: 10.3934/amc.2016044

Some new two-weight ternary and quinary codes of lengths six and twelve

1. 

University of Central Oklahoma, 100 North University Drive, P.O. Box 129, Edmond, OK 73034, United States

2. 

Colorado State University, 841 Oval Drive, P.O. Box 1874, Fort Collins, CO 80523-1874, United States

Received  February 2015 Published  November 2016

Let $[n,k]_q$ be a projective two-weight linear code over ${\rm GF}(q)^n$. In this correspondence, 9 codes are constructed in which $k=3$.
Citation: Liz Lane-Harvard, Tim Penttila. Some new two-weight ternary and quinary codes of lengths six and twelve. Advances in Mathematics of Communications, 2016, 10 (4) : 847-850. doi: 10.3934/amc.2016044
References:
[1]

A. S. Barlotti, $\{ k;n\}$-archi di un piano lineare finite, Boll. Un. Mat. Ital., 11 (1956), 553-556.

[2]

L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999), 211-213. doi: 10.1023/A:1008397209409.

[3]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243. doi: 10.1023/A:1005283806897.

[4]

R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā Indian J. Stat., 8 (1947), 107-166.

[5]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[6]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.

[7]

L. R. A Casse, W. A. Jackson, T. Penttila and G. F. Royle, Sets of type $(m,n)$ in $PG(2, r^2)$, $r$ odd,, in preparation., (). 

[8]

W. Cherowitzo, $\alpha$-flocks and hyperovals, Geom. Dedicata, 72 (1998), 221-246. doi: 10.1023/A:1005022808718.

[9]

W. Cherowitzo, C. M. O'Keefe and T. Penttila, A unified construction of finite geometries associated with q-clans in characteristic 2, Adv. Geom., 3 (2003), 1-21. doi: 10.1515/advg.2003.002.

[10]

W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.

[11]

F. De Clerck, S. De Winter and T. Maes, A geometric approach to Mathon maximal arcs, J. Combin. Theorey Ser. A, 118 (2001), 1196-1211. doi: 10.1016/j.jcta.2010.12.004.

[12]

F. De Clerck, S. De Winter and T. Maes, Partial flocks of the quadratic cone yielding Mathon maximal arcs, Discrete Math., 312 (2012), 2421-2428. doi: 10.1016/j.disc.2012.04.028.

[13]

F. De Clerck, S. De Winter and T. Maes, Singer 8-arcs of Mathon type in $PG(2,2^7)$, Des. Codes Cryptogr., 64 (2012), 17-31. doi: 10.1007/s10623-011-9502-4.

[14]

M. J. de Resmini, A 35-set of type $(2,5)$ in $PG(2,9)$, J. Combin. Theory Ser. A, 45 (1987), 303-305. doi: 10.1016/0097-3165(87)90021-5.

[15]

M. J. de Resmini and G. Migliori, A 78-set of type $(2,6)$ in $PG(2,16)$, Ars Combin., 22 (1986), 73-75.

[16]

The GAP Group, GAP - Groups, Algorithms, and Programming,, available online at , (). 

[17]

N. Hamilton, Maximal Arcs in Finite Projective Planes and Associated Structure in Projective Spaces, Ph.D. thesis, Univ. Western Australia, 1995.

[18]

N. Hamilton, Degree 8 maximal arcs in $PG(2,2^h)$, $h$ odd, J. Combin. Theory Ser. A, 100 (2002), 265-276. doi: 10.1006/jcta.2002.3297.

[19]

N. Hamilton and R. Mathon, More maximal arcs in Desarguesian projective planes and their geometric structure, Adv. Geom., 3 (2003), 251-261. doi: 10.1515/advg.2003.015.

[20]

N. Hamilton and R. Mathon, On the spectrum of non-Denniston maximal arcs in $PG(2,2h)$, Europ. J. Combin., 25 (2004), 415-421. doi: 10.1016/j.ejc.2003.07.006.

[21]

N. Hamilton and T. Penttila, Sets of type $(a,b)$ from subgroups of $\GammaL(1,p^R)$, J. Algebr. Combin., 13 (2001), 67-76. doi: 10.1023/A:1008775818040.

[22]

R. Mathon, New maximal arcs in Desarguesian planes, J. Combin. Theory Ser. A, 97 (2002), 353-368. doi: 10.1006/jcta.2001.3218.

[23]

C. M. O'Keefe and T. Penttila, A new hyperoval in $PG(2,32)$, J. Geom., 44 (1992), 117-139. doi: 10.1007/BF01228288.

[24]

S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128.

[25]

T. Penttila and G. F. Royle, Sets of type $(m,n)$ in the affine and projective planes of order nine, Des. Codes Cryptogr., 6 (1995), 229-245. doi: 10.1007/BF01388477.

[26]

B. Schmidt and C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17. doi: 10.1006/ffta.2000.0293.

[27]

B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finite, Ann. Mat. Pura Appl., 70 (1965), 1-201.

[28]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. I, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 812-818.

[29]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. II, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 1020-1025.

show all references

References:
[1]

A. S. Barlotti, $\{ k;n\}$-archi di un piano lineare finite, Boll. Un. Mat. Ital., 11 (1956), 553-556.

[2]

L. M. Batten and J. M. Dover, Some sets of type $(m,n)$ in cubic order planes, Des. Codes Cryptogr., 16 (1999), 211-213. doi: 10.1023/A:1008397209409.

[3]

A. Blokhuis and M. Lavrauw, Scattered spaces with respect to a spread in $PG(n,q)$, Geom. Dedicata, 81 (2000), 231-243. doi: 10.1023/A:1005283806897.

[4]

R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankhyā Indian J. Stat., 8 (1947), 107-166.

[5]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I. The user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[6]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.

[7]

L. R. A Casse, W. A. Jackson, T. Penttila and G. F. Royle, Sets of type $(m,n)$ in $PG(2, r^2)$, $r$ odd,, in preparation., (). 

[8]

W. Cherowitzo, $\alpha$-flocks and hyperovals, Geom. Dedicata, 72 (1998), 221-246. doi: 10.1023/A:1005022808718.

[9]

W. Cherowitzo, C. M. O'Keefe and T. Penttila, A unified construction of finite geometries associated with q-clans in characteristic 2, Adv. Geom., 3 (2003), 1-21. doi: 10.1515/advg.2003.002.

[10]

W. Cherowitzo, T. Penttila, I. Pinneri and G. F. Royle, Flocks and ovals, Geom. Dedicata, 60 (1996), 17-37. doi: 10.1007/BF00150865.

[11]

F. De Clerck, S. De Winter and T. Maes, A geometric approach to Mathon maximal arcs, J. Combin. Theorey Ser. A, 118 (2001), 1196-1211. doi: 10.1016/j.jcta.2010.12.004.

[12]

F. De Clerck, S. De Winter and T. Maes, Partial flocks of the quadratic cone yielding Mathon maximal arcs, Discrete Math., 312 (2012), 2421-2428. doi: 10.1016/j.disc.2012.04.028.

[13]

F. De Clerck, S. De Winter and T. Maes, Singer 8-arcs of Mathon type in $PG(2,2^7)$, Des. Codes Cryptogr., 64 (2012), 17-31. doi: 10.1007/s10623-011-9502-4.

[14]

M. J. de Resmini, A 35-set of type $(2,5)$ in $PG(2,9)$, J. Combin. Theory Ser. A, 45 (1987), 303-305. doi: 10.1016/0097-3165(87)90021-5.

[15]

M. J. de Resmini and G. Migliori, A 78-set of type $(2,6)$ in $PG(2,16)$, Ars Combin., 22 (1986), 73-75.

[16]

The GAP Group, GAP - Groups, Algorithms, and Programming,, available online at , (). 

[17]

N. Hamilton, Maximal Arcs in Finite Projective Planes and Associated Structure in Projective Spaces, Ph.D. thesis, Univ. Western Australia, 1995.

[18]

N. Hamilton, Degree 8 maximal arcs in $PG(2,2^h)$, $h$ odd, J. Combin. Theory Ser. A, 100 (2002), 265-276. doi: 10.1006/jcta.2002.3297.

[19]

N. Hamilton and R. Mathon, More maximal arcs in Desarguesian projective planes and their geometric structure, Adv. Geom., 3 (2003), 251-261. doi: 10.1515/advg.2003.015.

[20]

N. Hamilton and R. Mathon, On the spectrum of non-Denniston maximal arcs in $PG(2,2h)$, Europ. J. Combin., 25 (2004), 415-421. doi: 10.1016/j.ejc.2003.07.006.

[21]

N. Hamilton and T. Penttila, Sets of type $(a,b)$ from subgroups of $\GammaL(1,p^R)$, J. Algebr. Combin., 13 (2001), 67-76. doi: 10.1023/A:1008775818040.

[22]

R. Mathon, New maximal arcs in Desarguesian planes, J. Combin. Theory Ser. A, 97 (2002), 353-368. doi: 10.1006/jcta.2001.3218.

[23]

C. M. O'Keefe and T. Penttila, A new hyperoval in $PG(2,32)$, J. Geom., 44 (1992), 117-139. doi: 10.1007/BF01228288.

[24]

S. E. Payne, A new infinite family of generalized quadrangles, Congr. Numer., 49 (1985), 115-128.

[25]

T. Penttila and G. F. Royle, Sets of type $(m,n)$ in the affine and projective planes of order nine, Des. Codes Cryptogr., 6 (1995), 229-245. doi: 10.1007/BF01388477.

[26]

B. Schmidt and C. White, All two-weight irreducible cyclic codes?, Finite Fields Appl., 8 (2002), 1-17. doi: 10.1006/ffta.2000.0293.

[27]

B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finite, Ann. Mat. Pura Appl., 70 (1965), 1-201.

[28]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. I, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 812-818.

[29]

M. Tallini Scafati, $\{k,n\}$-archi di un pinao grafico finito, con particolare riguardo a quelli con due caratteri. II, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 40 (1966), 1020-1025.

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