The native conformations and thermodynamic properties of compact polymers with chain length N(N≤22) are investigated by using exact enumeration calculation method and Monte Carlo simulation method employing the HP model．The average Helmholtz free energy 〈F〉 and average partition function 〈Z〉 can be expressed in the form of〈F〉=aN+b

ln〈Z〉=a′N+b′

where N is chain length of compact chains a

b and a′

b′ only depend on temperature．We also find that the compact chains with unique ground states have similar relationship in the thermodynamic property and the chain length．In the HP model

the number of sequences of conformations for a protein of chain length N is 2N+1

and the number of sequences with unique ground state may be expressed as NS：NS=a×2N+1 (a=0.025)．In these HP sequences

the sequences with 40％～60％ in the percentage of Hresidue have a large probability of emergence．We find that many structures are the same in those unique ground state

and the values of degeneration of structure are 3.3～4.0 (10≤N≤16)．In the processing of folding

the change of energy is fast in the beginning of folding

slow in the middle of floding and also fast in the end of folding of proteins．