Table[PolynomialMod[α^k,α^3+α+1,Modulus->2],{k,0,7}]

Table[
{α^k,PolynomialMod[α^k,α^3+α+1,Modulus->2]},
{k,0,7}]//TableForm

<<FiniteFields` 

F16=GF[2,{1,1,0,0,1}];

Table[{k,b^k,ElementToPolynomial[Power[F16[{0,1,0,0}],k],b]//TraditionalForm,BaseForm[ElementToPolynomial[Power[F16[{0,1,0,0}],k],b]/. b->2,2],ElementToPolynomial[Power[F16[{0,1,0,0}],k],b]/. b->2},{k,1,15}]//TableForm

f[n_]:=(
n1=n;
vect={};
While[n1>0,
vect=Prepend[vect,Mod[n1,2]];
n1=Quotient[n1,2]];
Return[PadLeft[vect,4]];
)

f[5].Table[x^(3-k),{k,0,3}]

Expand[(1+x^2)*x^3]+PolynomialMod[(1+x^2)*x^3,x^3+x+1,
Modulus->2]

Table[{k,BaseForm[k,2],
Expand[f[k].{x^3,x^2,x,1}]  //TraditionalForm,
Expand[f[k].{x^3,x^2,x,1}*(x^3)]+PolynomialMod[f[k].{x^3,x^2,x,1}*(x^3),x^3+x+1,Modulus->2]//TraditionalForm},
{k,0,15}]//TableForm

Table[{k,BaseForm[k,2],
Expand[f[k].{x^3,x^2,x,1}]  //TraditionalForm,
Expand[f[k].{x^3,x^2,x,1}*(x^3)]+PolynomialMod[f[k].{x^3,x^2,x,1}*(x^3),x^3+x+1,Modulus->2]//TraditionalForm,
PadLeft[Reverse[CoefficientList[
Expand[f[k].{x^3,x^2,x,1}*(x^3)]+
PolynomialMod[f[k].{x^3,x^2,x,1}*(x^3),x^3+x+1,Modulus->2],x]
],7]//StandardForm},
{k,0,15}]//TableForm

words=Table[
PadLeft[Reverse[CoefficientList[
Expand[f[k].{x^3,x^2,x,1}*(x^3)]
+PolynomialMod[f[k].{x^3,x^2,x,1}*(x^3),x^3+x+1,
Modulus->2],x]],7],
{k,0,15}];
Table[Table[HammingDistance[words[[i]],words[[j]]],
{j,1,16}],{i,1,16}]