After ordering any such improvements, as provided in sections 31-727 to 31-762, the board of trustees or the administrator may enter into a contract for the construction of such improvement in one or more contracts, but no work shall be done or contract let until notice to contractors has been published in a legal newspaper of general circulation in the county where the district is organized. The notice shall be published the same day each week two consecutive weeks in such paper and shall generally state (1) the extent of the work, (2) the kinds of material to be bid upon, including in such notice all kinds of material mentioned in the resolution as provided in section 31-744, (3) the amount of the engineer's estimate of the cost of such improvements, (4) the time when bids will be received, and (5) the amount of the certified check or bid bond required to accompany the bids. Each bid shall be accompanied in a separate sealed envelope by certified check or bid bond in an amount to be named in the notice, which amount shall be not less than five percent of the engineer's total estimate of the cost, and shall be made payable to the treasurer of the district as security that the bidder to whom the contract may be awarded will enter into a contract to build the improvements in accordance with the notice to contractors and give bond in the sum named in such notice for the construction of such improvements as the notice required. Checks or bonds accompanying bids not accepted shall be returned to the bidders. The work provided for in this section shall be done under written contract with the lowest responsible bidder on the material selected after the bids are opened and in accordance with the requirements of the plans and specifications. The board of trustees or the administrator may reject any or all bids received and advertise for new bids in accordance with this section.