How to design a balanced LFSR with period 1023 and linear span 55? I am trying various combination of LFSR's to try to design sequence of period 1023.
The linear span should be 55 and the output 1023 bits has to be balanced.

I have tried with five LFSR's with degree 10 and one LFSR with degree 5, XORing all ..which gives 55 as Linear span and period 1023.
However, it is not balanced.
With trial and error, I get a balanced output with XORing 9 LFSR of degree 10 and one LFSR of deg 5. 

Can anyone give some idea what path to take to solve this ?
The polynomial factorisation of $X^{2^L-1}+1$ into irreducible factors gives you all the polynomials $g_i(X)$ that can be used as LFSR polynomials in generating any sequence of period $2^L-1.$ Say your goal is to generate a sequence of this period with linear complexity $c.$ Assume

$$(X^{2^L-1}+1)=\prod_{i=1}^v g_i(X),$$
there will be no repeated factors. Not all factors will be primitive of course.

Then for each $g_i(X)|(X^{2^L-1}+1),$ find the linear complexity of the periodic sequence it generates, say it is $L_i.$ You can use Berlekamp Massey for this.

If you can find a subset $J\subset \{1,2,\ldots,v\}$ with $\sum_{i\in J}L_i=c,$ then the corresponding polynomials $$g_i(X),\quad i\in J$$ will generate the sequence you need, for some initial conditions. Some of these polynomials will generate a sequence of length dividing $2^L-1$ of course.

More generally, given polynomials $f_i$ and letting $\Omega(f_i)$ be the set of sequences they generate, then 
$$
\Omega(f_1)+\cdots+\Omega(f_v)=\Omega(lcm(f_1,\ldots,f_v)),
$$
and this has been known since the 1950s, published by Zierler in 1959, in Siam Journal of Discrete Maths. 

Another good and accessible but somewhat dated reference is the book by Rueppel, The Analysis and Design of Stream Ciphers. 

Appendix:

Magma online calculator gives the following factors for $X^{1023}+1$ (truncated)


  X + 1, X^2 + X + 1, X^5 + X^2 + 1, 1>, other degree 5 factors, 
  X^10 + X^3 + X^2 + X + 1, other degree 10 factors


so what you found experimentally is not surprising.

Edit: As I alluded to, and this holds for any characteristic, but I will state it for binary, given any sequence $s(t)$ over $F_2$ with period dividing $2^L-1$ (and thus for period equal to $2^L-1$) there is an appropriate trace expansion for that sequence of the form
$$
s(t)=\sum_{i \in J} tr^{L_i}_1(\theta_i \alpha^{i t})
$$
and each of those trace sequences are some cyclic shift of some maximal length sequence (equivalently some cyclic shift of the output of some primitive LFSR) with length $L_i | L,$ where $L_i$ is the number of elements in the cyclotomic coset of $i$ modulo $2^L-1.$ So if a sequence with the parameters you want exists (w.r.t. linear complexity) it can be constructed this way.