Publix Super Markets (PUSH) 10-Q 2010 Q1 Quarterly report

Exhibit 32.1

CERTIFICATION PURSUANT TO 18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

The certification set forth below is being submitted in connection with the Quarterly Report of Publix Super Markets, Inc. (the “Company”) on Form 10-Q for the period ending March 27, 2010 (the “Report”) for the purpose of complying with Rule 13a-14(b) or Rule 15d-14(b) of the Securities Exchange Act of 1934 (the “Exchange Act”) and Section 1350 of Chapter 63 of Title 18 of the United States Code.

I, William E. Crenshaw, Chief Executive Officer of the Company, certify, to the best of my knowledge, that on the date hereof:

(1) The Report fully complies with the requirements of Section 13(a) or 15(d) of the Exchange Act; and

(2) The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

/s/ William E. Crenshaw

William E. Crenshaw

Chief Executive Officer

May 6, 2010