61,881 research outputs found
Real Closed Exponential Subfields of Pseudoexponential Fields
In this paper, we prove that a pseudoexponential field has continuum many
non-isomorphic countable real closed exponential subfields, each with an order
preserving exponential map which is surjective onto the nonnegative elements.
Indeed, this is true of any algebraically closed exponential field satisfying
Schanuel's conjecture
Construction of self-dual normal bases and their complexity
Recent work of Pickett has given a construction of self-dual normal bases for
extensions of finite fields, whenever they exist. In this article we present
these results in an explicit and constructive manner and apply them, through
computer search, to identify the lowest complexity of self-dual normal bases
for extensions of low degree. Comparisons to similar searches amongst normal
bases show that the lowest complexity is often achieved from a self-dual normal
basis
Explicit Space-Time Codes Achieving The Diversity-Multiplexing Gain Tradeoff
A recent result of Zheng and Tse states that over a quasi-static channel,
there exists a fundamental tradeoff, referred to as the diversity-multiplexing
gain (D-MG) tradeoff, between the spatial multiplexing gain and the diversity
gain that can be simultaneously achieved by a space-time (ST) block code. This
tradeoff is precisely known in the case of i.i.d. Rayleigh-fading, for T>=
n_t+n_r-1 where T is the number of time slots over which coding takes place and
n_t,n_r are the number of transmit and receive antennas respectively. For T <
n_t+n_r-1, only upper and lower bounds on the D-MG tradeoff are available.
In this paper, we present a complete solution to the problem of explicitly
constructing D-MG optimal ST codes, i.e., codes that achieve the D-MG tradeoff
for any number of receive antennas. We do this by showing that for the square
minimum-delay case when T=n_t=n, cyclic-division-algebra (CDA) based ST codes
having the non-vanishing determinant property are D-MG optimal. While
constructions of such codes were previously known for restricted values of n,
we provide here a construction for such codes that is valid for all n.
For the rectangular, T > n_t case, we present two general techniques for
building D-MG-optimal rectangular ST codes from their square counterparts. A
byproduct of our results establishes that the D-MG tradeoff for all T>= n_t is
the same as that previously known to hold for T >= n_t + n_r -1.Comment: Revised submission to IEEE Transactions on Information Theor
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