ENEE626 (2008). Problem set 4. Due in class on Nov. 25.
1. (10 pt)
(a) Let C⊂ {0,1}nbe a linear binary code. Let h1= (000),h2= (001),...,h8= (111).Let
As1,...,s8(C)be the number of ordered triples of linearly independent vectors of Csuch that
|{i∈[1..n]:(c1,i, c2,i, c3,i ) = hj}| =sj,1≤j≤8.
Prove that there exist binary linear codes such that for any 8-tuple of nonnegative integers s1, . . . , s8where
s1+· · · +s8=n,
As1,...,s8(C)≤n82−3(n−k)n
s1, . . . , s8.
(b) Generalize this claim from triples to t-tuples.
2. (20 pt) Let S={x∈Fn
2: wt(x) = w}.
(a) Suppose that vectors from Sare chosen with uniform distribution. Let Xibe the random variable
whose value equals the ith coordinate of the vector. What is P(Xi= 1)?
(b) Given n, i, j, are Xiand Xjindependent r.v.’s? What if n→ ∞?What about r.v.’s Xi1, Xi2, . . . , Xir?
Consider the cases of rfixed, r=o(n),and r= (const)n.
(c) What is the typical distance between x,y∈S? In other words, what is αsuch that
lim
n→∞,w→∞
w/n→ω
0<ω<1
P(|d(x,y)−αn| ≥ n) = 0 (∀ > 0) ?
State and prove a precise claim that answers this question.
(d) Given 4 randomly chosen vectors of weight w, what is the typical number of coordinates iin which
(X1,i, X2,i, X3,i , X4,i) = (0,0,0,0)? (1,1,1,1)? (1,0,0,0)? (1,1,0,0)?
3. (10pt) Solve Problem 5 from final exam of 2007 (posted on class web page).
4. (10 pt) Let R(C)be the code rate and let δ(C)be the relative distance of a given code C. Define the
functions
Rm(δ) = lim sup
n→∞
max{R(C)|δ(C)≥δ, C ⊂Fn
2}
δm(R) = lim sup
n→∞
max{δ(C)|R(C)≥R, C ⊂Fn
2}.
Let δm(x)≤f(x)for a continuous, strictly decreasing function fdefined on an open interval. Prove that
then Rm≤f−1(δ).
5. (10 pt) Consider the ensemble of linear codes given by randomly chosen (n−k)×nparity-check
matrices.
(a) Let D(C)be the r.v. equal to the distance of a random code from the ensemble. Find limn→∞
ED(C)
n.
(b) What is the typical relative distance of codes in this ensemble? (Formulate a precise claim and prove
it.)
6. (10 pt) Let n→ ∞, m =µn, r =ρn, s =σn; 0 < ρ < µ.
(a) Compute the exponential asymptotics of
F(r) = s
rn−s
m−r.
In other words, find αsuch that
lim
n→∞
1
nlog2
F(r)
2αn = 0.
