We use a theorem of Chow (1949) on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question
of Beelen et al. (2010) concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chows theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians.
In this paper we consider the problem of determining the weight spectrum of q-ary codes C(3,m) associated with Grassmann varieties G(3,m). For m=6 this was done by Nogin. We derive a formula for the weight of a codeword of C(3,m), in terms of certain
varieties associated with alternating trilinear forms on (F_q)^m. The classification of such forms under the action of the general linear group GL(m,F_q) is the other component that is required to calculate the spectrum of C(3,m). For m=7, we explicitly determine the varieties mentioned above. The classification problem for alternating 3-forms on (F_q)^7 was solved by Cohen and Helminck, which we then use to determine the spectrum of C(3,7).
In this paper, we introduce the so-called multiscale limit for spectral curves, associated with real finite-gap Sine-Gordon solutions. This technique allows to solve the old problem of calculating the density of topological charge for real finite-gap
Sine-Gordon solutions directly from the $theta$-functional formulas.
The most basic characteristic of x-quasiperiodic solutions u(x,t) of the sine-Gordon equation u_{tt}-u_{xx}+sin u=0 is the topological charge density denoted $bar n$. The real finite-gap solutions u(x,t) are expressed in terms of the Riemann theta-fu
nctions of a non-singular hyperelliptic curve $Gamma$ and a positive generic divisor D of degree g on $Gamma$, where the spectral data $(Gamma, D)$ must satisfy some reality conditions. The problem addressed in note is: to calculate $bar n$ directly from the theta-functional expressions for the solution u(x,t). The problem is solved here by introducing what we call the multiscale or elliptic limit of real finite-gap sine-Gordon solutions. We deform the spectral curve to a singular curve, for which the calculation of topological charges reduces to two special easier cases.
