# Shaurya Baranwal Homework 9 due February 18, 2024

#--------------------------------------
#Important functions for this homework|
#--------------------------------------

Fqn:=proc(q,n) local S,a,v:

  option remember:
  if n=0 then
    RETURN({[]}):
  fi:

  S:=Fqn(q,n-1):

  {seq(seq([op(v),a],a=0..q-1), v in S)}:

end:

Syn:=proc(q,H,y) local i:

  [seq(DP(q,y,H[i]),i=1..nops(H))]:

end:

DP:=proc(q,u,v) local i,n:

  n:=nops(u):
  add(u[i]*v[i],i=1..n) mod q:

end:

SynT:=proc(q,M) local n,T,A,H,S,s,i:

  option remember:

  if SFde(q,M)<>M then
    RETURN(FAIL):
  fi:

  n:=nops(M[1]):
  A:=SAah(q,n,M):
  H:=PCM(q,M):
  S:={}:

  for i from 1 to nops(A) do
    s:=Syn(q,H,A[i][1]):
    S:=S union {s}:
    T[s]:=A[i][1]:
  od:

  op(T),S:

end:

PCM:=proc(q,M) local k,n,i,j,H:

  option remember:
  k:=nops(M):
  n:=nops(M[1]):

  if [seq([op(1..k,M[i])],i=1..k)]<>[seq([0$(i-1),1,0$(k-i)],i=1..k)] then
    print(`Not standard form , please use SFde(q,M) first `):
    RETURN(FAIL):
  fi:

  for i from 1 to n-k do
    for j from 1 to k do
      H[i,j]:=-M[j][i+k]  mod q:
    od:
    for j from k+1 to n do
      if j-k=i then
        H[i,j]:=1:
      else
        H[i,j]:=0:
      fi:
    od:
  od:

  [seq([seq(H[i,j],j=1..n)],i=1..n-k)]:

end:

#--------------------------------------
# Part 1 - Finding out everyone's age |
#--------------------------------------

#AjBu N/A 
#AuroraHiveley 23 
#AlexValentino N/A
#AlexVarjabedian 19
#DanielElwell N/A
#DayoonKim N/A
#GeorgeSpahn N/A
#GloriaLiu 21
#HimanshuChandra 49
#IsaacLam 18
#JosephKoutsoutis 20
#KayleeWeatherspoon 22
#LucyMartinez 25
#NatalyaTer-Saakov 24
#NkhaloMalawo N/A
#NurayKutlu 20
#PabloBlanco 22
#RameshBalaji 19
#RDB (aka Robert Dougherty-Bliss) N/A
#RyanBadi 20
#ShauryaBaranwal 19

#--------------------------------------
#       Part 2 - antiPCM(q,H)         |
#--------------------------------------

AntiPCM := proc(q, H) local n, k, S, i, j:

  k := nops(H):
  n := nops(H[1]):
  S := [[0$(n-k)]$n]:
  
  for i from 1 to n-k do:
    for j from 1 to n-k do:
      if i=j then S[i][j] := 1 else S[i][j] := 0 fi:
    od:
  od:
  
  for i from 1 to k do:
    for j from 1 to n-k do:
      S[j][k+i+1] := -H[i][j] mod q:
    od:
  od:

  return S:

end:

#--------------------------------------
#         Part 3 - DecodeLT           |
#--------------------------------------

DecodeLT := proc(q, M) local n, fqn, H, v, L:

  n := nops(M[1]):
  fqn := Fqn(q, n):
  H := AntiPCM(q, M):

  for v in fqn do:
    L[v] := Syn(q, H, v):
  od:

  return L:

end: